% This TeX file was generated by texpsinclude
% from the original TeX file powerlawrev.tex
% and the PostScript files expcor01.eps expcor02.eps.
% TeX writes out the included PostScript to files.
%
% Here is part of powerlawrev.tex:
%powerlawrev.tex

\documentclass[12pt]{amsart}

% Here are the
% TeX macros for dumping included Postscript to files.
% Adapted from Knuth's \answer macro in the TeXbook.
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94

\def\endofps{EndOfTheIncludedPostscriptMagicCookie}
\chardef\other=12
\newwrite\psdumphandle 
\outer\def\psdump#1{\par\medbreak
  \immediate\openout\psdumphandle=#1
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copypsline}
\def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials
  \catcode`\\=\other \obeylines}
{\obeylines \gdef\copypsline#1
  {\def\next{#1}%
  \ifx\next\endofps\let\next=\endgroup %
  \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}}
\outer\def\closepsdump{
  \immediate\closeout\psdumphandle}

% Here is the PostScript for expcor01.eps:
\message{Writing file expcor01.eps}
\psdump{expcor01.eps}%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(TM) 5.5
%%For: (Kenneth Alexander) (USC)
%%Title: (expcor01B.eps)
%%CreationDate: (5/23/2000) (1:32 PM)
%%BoundingBox: 63 346 543 715
%%HiResBoundingBox: 63 346.7689 543 714.1327
%%DocumentProcessColors: Black
%%DocumentFonts: Symbol
%%+ Times-Roman
%%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0
%%+ procset Adobe_typography_AI5 1.0 0
%%+ procset Adobe_IllustratorA_AI5 1.0 0
%AI5_FileFormat 1.2
%AI3_ColorUsage: Black&White
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 30 31 582 761
%AI3_DocumentPreview: Header
%AI5_ArtSize: 612 792
%AI5_RulerUnits: 2
%AI5_ArtFlags: 1 0 0 1 0 0 1 1 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 1
%AI5_OpenToView: -54 828 -1 693 507 18 0 1 75 50
%AI5_OpenViewLayers: 7
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_level2_AI5 1.0 0
%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.0 
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 21 dict dup begin
	put
	/packedarray where not
	{
		userdict begin
		/packedarray
		{
			array astore readonly
		} bind def
		/setpacking /pop load def
		/currentpacking false def
	 end
		0
	} if
	pop
	userdict /defaultpacking currentpacking put true setpacking
	/initialize
	{
		Adobe_level2_AI5 begin
	} bind def
	/terminate
	{
		currentdict Adobe_level2_AI5 eq
		{
		 end
		} if
	} bind def
	mark
	/setcustomcolor where not
	{
		/findcmykcustomcolor
		{
			5 packedarray
		} bind def
		/setcustomcolor
		{
			exch aload pop pop
			4
			{
				4 index mul 4 1 roll
			} repeat
			5 -1 roll pop
			setcmykcolor
		}
		def
	} if
	
	/gt38? mark {version cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def
	userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put
	userdict /level2?
	systemdict /languagelevel known dup
	{
		pop systemdict /languagelevel get 2 ge
	} if
	put
	level2? not
	{
		/setcmykcolor where not
		{
			/setcmykcolor
			{
				exch .11 mul add exch .59 mul add exch .3 mul add
				1 exch sub setgray
			} def
		} if
		/currentcmykcolor where not
		{
			/currentcmykcolor
			{
				0 0 0 1 currentgray sub
			} def
		} if
		/setoverprint where not
		{
			/setoverprint /pop load def
		} if
		/selectfont where not
		{
			/selectfont
			{
				exch findfont exch
				dup type /arraytype eq
				{
					makefont
				}
				{
					scalefont
				} ifelse
				setfont
			} bind def
		} if
		/cshow where not
		{
			/cshow
			{
				[
				0 0 5 -1 roll aload pop
				] cvx bind forall
			} bind def
		} if
	} if
	cleartomark
	/anyColor?
	{
		add add add 0 ne
	} bind def
	/testColor
	{
		gsave
		setcmykcolor currentcmykcolor
		grestore
	} bind def
	/testCMYKColorThrough
	{
		testColor anyColor?
	} bind def
	userdict /composite?
	level2?
	{
		gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore
		add add add 4 eq
	}
	{
		1 0 0 0 testCMYKColorThrough
		0 1 0 0 testCMYKColorThrough
		0 0 1 0 testCMYKColorThrough
		0 0 0 1 testCMYKColorThrough
		and and and
	} ifelse
	put
	composite? not
	{
		userdict begin
		gsave
		/cyan? 1 0 0 0 testCMYKColorThrough def
		/magenta? 0 1 0 0 testCMYKColorThrough def
		/yellow? 0 0 1 0 testCMYKColorThrough def
		/black? 0 0 0 1 testCMYKColorThrough def
		grestore
		/isCMYKSep? cyan? magenta? yellow? black? or or or def
		/customColor? isCMYKSep? not def
	 end
	} if
 end defaultpacking setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0 
%%CreationDate:(03/26/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 54 dict dup begin
put
/initialize
{
 begin
 begin
	Adobe_typography_AI5 begin
	Adobe_typography_AI5
	{
		dup xcheck
		{
			bind
		} if
		pop pop
	} forall
 end
 end
 end
	Adobe_typography_AI5 begin
} def
/terminate
{
	currentdict Adobe_typography_AI5 eq
	{
	 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	/_pntr 0 ddef
	{
		counttomark -1 roll
		dup type dup /marktype eq
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef
			} ifelse
		} ifelse
	} loop
	_tempEncode
} def
/TE
{
	StandardEncoding 256 array copy modifyEncoding
	/_nativeEncoding exch def
} def
%
/TZ
{
	dup type /arraytype eq
	{
		/_wv exch def
	}
	{
		/_wv 0 def
	} ifelse
	/_useNativeEncoding exch def
	pop pop
	findfont _wv type /arraytype eq
	{
		_wv makeblendedfont
	} if
	dup length 2 add dict
 begin
	mark exch
	{
		1 index /FID ne
		{
			def
		} if
		cleartomark mark
	} forall
	pop
	/FontName exch def
	counttomark 0 eq
	{
		1 _useNativeEncoding eq
		{
			/Encoding _nativeEncoding def
		} if
		cleartomark
	}
	{
		/Encoding load 256 array copy
		modifyEncoding /Encoding exch def
	} ifelse
	FontName currentdict
 end
	definefont pop
} def
/tr
{
	_ax _ay 3 2 roll
} def
/trj
{
	_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
	/Tx
	{
		dup
		currentpoint 3 2 roll
		tr _psf
		newpath moveto
		tr _ctm _pss
	} ddef
	/Tj
	{
		dup
		currentpoint 3 2 roll
		trj _pjsf
		newpath moveto
		trj _ctm _pjss
	} ddef
} def
/a1
{
	/Tx
	{
		dup currentpoint 4 2 roll gsave
		dup currentpoint 3 2 roll
		tr _psf
		newpath moveto
		tr _ctm _pss
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll gsave
		dup currentpoint 3 2 roll
		trj _pjsf
		newpath moveto
		trj _ctm _pjss
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/e0
{
	/Tx
	{
		tr _psf
	} ddef
	/Tj
	{
		trj _pjsf
	} ddef
} def
/e1
{
	/Tx
	{
		dup currentpoint 4 2 roll gsave
		tr _psf
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll gsave
		trj _pjsf
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/i0
{
	/Tx
	{
		tr sp
	} ddef
	/Tj
	{
		trj jsp
	} ddef
} def
/i1
{
	W N
} def
/o0
{
	/Tx
	{
		tr sw rmoveto
	} ddef
	/Tj
	{
		trj swj rmoveto
	} ddef
} def
/r0
{
	/Tx
	{
		tr _ctm _pss
	} ddef
	/Tj
	{
		trj _ctm _pjss
	} ddef
} def
/r1
{
	/Tx
	{
		dup currentpoint 4 2 roll currentpoint gsave newpath moveto
		tr _ctm _pss
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll currentpoint gsave newpath moveto
		trj _ctm _pjss
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix
	_tDict begin
	/W
	{
	} def
	/h
	{
	} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le
	{
		currentpoint newpath moveto
	} if
	dup 8 eq
	{
		pop 0
	}
	{
		dup 9 eq
		{
			pop 1
		} if
	} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
	_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
	_tm astore pop iTm 0 0 moveto
} def
/Td
{
	_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq
	{
	}
	{
		_renderEnd _render get dup null ne
		{
			load exec
		}
		{
			pop
		} ifelse
	} ifelse
	/_render -1 ddef
} def
/Ta
{
	pop
} def
/Tf
{
	dup 1000 div /_fScl exch ddef
%
	selectfont
} def
/Tl
{
	pop
	0 exch _leading astore pop
} def
/Tt
{
	pop
} def
/TW
{
	3 npop
} def
/Tw
{
	/_cx exch ddef
} def
/TC
{
	3 npop
} def
/Tc
{
	/_ax exch ddef
} def
/Ts
{
	/_rise exch ddef
	currentpoint
	iTm
	moveto
} def
/Ti
{
	3 npop
} def
/Tz
{
	100 div /_hs exch ddef
	iTm
} def
/TA
{
	pop
} def
/Tq
{
	pop
} def
/Th
{
	pop pop pop pop pop
} def
/TX
{
	pop
} def
/Tk
{
	exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
	2 npop
} def
/T*
{
	_leading aload pop neg Td
} def
/T*-
{
	_leading aload pop Td
} def
/T-
{
	_hyphen Tx
} def
/T+
{
} def
/TR
{
	_ctm currentmatrix pop
	_tm astore pop
	iTm 0 0 moveto
} def
/TS
{
	currentfont 3 1 roll
	/_Symbol_ _fScl 1000 mul selectfont
	
	0 eq
	{
		Tx
	}
	{
		Tj
	} ifelse
	setfont
} def
/Xb
{
	pop pop
} def
/Tb /Xb load def
/Xe
{
	pop pop pop pop
} def
/Te /Xe load def
/XB
{
} def
/TB /XB load def
currentdict readonly pop
end
setpacking
%%EndResource
%%BeginResource: procset Adobe_IllustratorA_AI5 1.1 0
%%Title: (Adobe Illustrator (R) Version 5.0 Abbreviated Prolog)
%%Version: 1.1 
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_IllustratorA_AI5_vars 70 dict dup begin
put
/_lp /none def
/_pf
{
} def
/_ps
{
} def
/_psf
{
} def
/_pss
{
} def
/_pjsf
{
} def
/_pjss
{
} def
/_pola 0 def
/_doClip 0 def
/cf currentflat def
/_tm matrix def
/_renderStart
[
/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0
] def
/_renderEnd
[
null null null null /i1 /i1 /i1 /i1
] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading
[
0 0
] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc
{
} def
/discardSave null def
/buffer 256 string def
/beginString null def
/endString null def
/endStringLength null def
/layerCnt 1 def
/layerCount 1 def
/perCent (%) 0 get def
/perCentSeen? false def
/newBuff null def
/newBuffButFirst null def
/newBuffLast null def
/clipForward? false def
end
userdict /Adobe_IllustratorA_AI5 74 dict dup begin
put
/initialize
{
	Adobe_IllustratorA_AI5 dup begin
	Adobe_IllustratorA_AI5_vars begin
	discardDict
	{
		bind pop pop
	} forall
	dup /nc get begin
	{
		dup xcheck 1 index type /operatortype ne and
		{
			bind
		} if
		pop pop
	} forall
 end
	newpath
} def
/terminate
{
 end
 end
} def
/_
null def
/ddef
{
	Adobe_IllustratorA_AI5_vars 3 1 roll put
} def
/xput
{
	dup load dup length exch maxlength eq
	{
		dup dup load dup
		length 2 mul dict copy def
	} if
	load begin
	def
 end
} def
/npop
{
	{
		pop
	} repeat
} def
/sw
{
	dup length exch stringwidth
	exch 5 -1 roll 3 index mul add
	4 1 roll 3 1 roll mul add
} def
/swj
{
	dup 4 1 roll
	dup length exch stringwidth
	exch 5 -1 roll 3 index mul add
	4 1 roll 3 1 roll mul add
	6 2 roll /_cnt 0 ddef
	{
		1 index eq
		{
			/_cnt _cnt 1 add ddef
		} if
	} forall
	pop
	exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
	4 1 roll
	{
		2 npop
		(0) exch 2 copy 0 exch put pop
		gsave
		false charpath currentpoint
		4 index setmatrix
		stroke
		grestore
		moveto
		2 copy rmoveto
	} exch cshow
	3 npop
} def
/jss
{
	4 1 roll
	{
		2 npop
		(0) exch 2 copy 0 exch put
		gsave
		_sp eq
		{
			exch 6 index 6 index 6 index 5 -1 roll widthshow
			currentpoint
		}
		{
			false charpath currentpoint
			4 index setmatrix stroke
		} ifelse
		grestore
		moveto
		2 copy rmoveto
	} exch cshow
	6 npop
} def
/sp
{
	{
		2 npop (0) exch
		2 copy 0 exch put pop
		false charpath
		2 copy rmoveto
	} exch cshow
	2 npop
} def
/jsp
{
	{
		2 npop
		(0) exch 2 copy 0 exch put
		_sp eq
		{
			exch 5 index 5 index 5 index 5 -1 roll widthshow
		}
		{
			false charpath
		} ifelse
		2 copy rmoveto
	} exch cshow
	5 npop
} def
/pl
{
	transform
	0.25 sub round 0.25 add exch
	0.25 sub round 0.25 add exch
	itransform
} def
/setstrokeadjust where
{
	pop true setstrokeadjust
	/c
	{
		curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y
	{
		2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		lineto
	} def
	/L
	/l load def
	/m
	{
		moveto
	} def
}
{
	/c
	{
		pl curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll pl curveto
	} def
	/V
	/v load def
	/y
	{
		pl 2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		pl lineto
	} def
	/L
	/l load def
	/m
	{
		pl moveto
	} def
} ifelse
/d
{
	setdash
} def
/cf
{
} def
/i
{
	dup 0 eq
	{
		pop cf
	} if
	setflat
} def
/j
{
	setlinejoin
} def
/J
{
	setlinecap
} def
/M
{
	setmiterlimit
} def
/w
{
	setlinewidth
} def
/H
{
} def
/h
{
	closepath
} def
/N
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			clip /_doClip 0 ddef
		} if
		newpath
	}
	{
		/CRender
		{
			N
		} ddef
	} ifelse
} def
/n
{
	N
} def
/F
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _pf grestore clip newpath /_lp /none ddef _fc
			/_doClip 0 ddef
		}
		{
			_pf
		} ifelse
	}
	{
		/CRender
		{
			F
		} ddef
	} ifelse
} def
/f
{
	closepath
	F
} def
/S
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _ps grestore clip newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			_ps
		} ifelse
	}
	{
		/CRender
		{
			S
		} ddef
	} ifelse
} def
/s
{
	closepath
	S
} def
/B
{
	_pola 0 eq
	{
		_doClip 1 eq
		gsave F grestore
		{
			gsave S grestore clip newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			S
		} ifelse
	}
	{
		/CRender
		{
			B
		} ddef
	} ifelse
} def
/b
{
	closepath
	B
} def
/W
{
	/_doClip 1 ddef
} def
/*
{
	count 0 ne
	{
		dup type /stringtype eq
		{
			pop
		} if
	} if
	newpath
} def
/u
{
} def
/U
{
} def
/q
{
	_pola 0 eq
	{
		gsave
	} if
} def
/Q
{
	_pola 0 eq
	{
		grestore
	} if
} def
/*u
{
	_pola 1 add /_pola exch ddef
} def
/*U
{
	_pola 1 sub /_pola exch ddef
	_pola 0 eq
	{
		CRender
	} if
} def
/D
{
	pop
} def
/*w
{
} def
/*W
{
} def
/`
{
	/_i save ddef
	clipForward?
	{
		nulldevice
	} if
	6 1 roll 4 npop
	concat pop
	userdict begin
	/showpage
	{
	} def
	0 setgray
	0 setlinecap
	1 setlinewidth
	0 setlinejoin
	10 setmiterlimit
	[] 0 setdash
	/setstrokeadjust where {pop false setstrokeadjust} if
	newpath
	0 setgray
	false setoverprint
} def
/~
{
 end
	_i restore
} def
/O
{
	0 ne
	/_of exch ddef
	/_lp /none ddef
} def
/R
{
	0 ne
	/_os exch ddef
	/_lp /none ddef
} def
/g
{
	/_gf exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_gf setgray
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/G
{
	/_gs exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_gs setgray
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/k
{
	_cf astore pop
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_cf aload pop setcmykcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/K
{
	_cs astore pop
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_cs aload pop setcmykcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/x
{
	/_gf exch ddef
	findcmykcustomcolor
	/_if exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_if _gf 1 exch sub setcustomcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/X
{
	/_gs exch ddef
	findcmykcustomcolor
	/_is exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_is _gs 1 exch sub setcustomcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/A
{
	pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/discard
{
	save /discardSave exch store
	discardDict begin
	/endString exch store
	gt38?
	{
		2 add
	} if
	load
	stopped
	pop
 end
	discardSave restore
} bind def
userdict /discardDict 7 dict dup begin
put
/pre38Initialize
{
	/endStringLength endString length store
	/newBuff buffer 0 endStringLength getinterval store
	/newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store
	/newBuffLast newBuff endStringLength 1 sub 1 getinterval store
} def
/shiftBuffer
{
	newBuff 0 newBuffButFirst putinterval
	newBuffLast 0
	currentfile read not
	{
	stop
	} if
	put
} def
0
{
	pre38Initialize
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff endString eq
			{
				cleartomark stop
			} if
			shiftBuffer
		} loop
	}
	{
	stop
	} ifelse
} def
1
{
	pre38Initialize
	/beginString exch store
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff beginString eq
			{
				/layerCount dup load 1 add store
			}
			{
				newBuff endString eq
				{
					/layerCount dup load 1 sub store
					layerCount 0 eq
					{
						cleartomark stop
					} if
				} if
			} ifelse
			shiftBuffer
		} loop
	}
	{
	stop
	} ifelse
} def
2
{
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		endString eq
		{
			cleartomark stop
		} if
	} loop
} def
3
{
	/beginString exch store
	/layerCnt 1 store
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		dup beginString eq
		{
			pop /layerCnt dup load 1 add store
		}
		{
			endString eq
			{
				layerCnt 1 eq
				{
					cleartomark stop
				}
				{
					/layerCnt dup load 1 sub store
				} ifelse
			} if
		} ifelse
	} loop
} def
end
userdict /clipRenderOff 15 dict dup begin
put
{
	/n /N /s /S /f /F /b /B
}
{
	{
		_doClip 1 eq
		{
			/_doClip 0 ddef clip
		} if
		newpath
	} def
} forall
/Tr /pop load def
/Bb {} def
/BB /pop load def
/Bg {12 npop} def
/Bm {6 npop} def
/Bc /Bm load def
/Bh {4 npop} def
end
/Lb
{
	4 npop
	6 1 roll
	pop
	4 1 roll
	pop pop pop
	0 eq
	{
		0 eq
		{
			(%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard
		}
		{
			/clipForward? true def
			
			/Tx /pop load def
			/Tj /pop load def
			currentdict end clipRenderOff begin begin
		} ifelse
	}
	{
		0 eq
		{
			save /discardSave exch store
		} if
	} ifelse
} bind def
/LB
{
	discardSave dup null ne
	{
		restore
	}
	{
		pop
		clipForward?
		{
			currentdict
		 end
		 end
		 begin
			
			/clipForward? false ddef
		} if
	} ifelse
} bind def
/Pb
{
	pop pop
	0 (%AI5_EndPalette) discard
} bind def
/Np
{
	0 (%AI5_End_NonPrinting--) discard
} bind def
/Ln /pop load def
/Ap
/pop load def
/Ar
{
	72 exch div
	0 dtransform dup mul exch dup mul add sqrt
	dup 1 lt
	{
		pop 1
	} if
	setflat
} def
/Mb
{
	q
} def
/Md
{
} def
/MB
{
	Q
} def
/nc 3 dict def
nc begin
/setgray
{
	pop
} bind def
/setcmykcolor
{
	4 npop
} bind def
/setcustomcolor
{
	2 npop
} bind def
currentdict readonly pop
end
currentdict readonly pop
end
setpacking
%%EndResource
%%EndProlog
%%BeginSetup
%%IncludeFont: Symbol
%%IncludeFont: Times-Roman
Adobe_level2_AI5 /initialize get exec
Adobe_IllustratorA_AI5_vars Adobe_IllustratorA_AI5 Adobe_typography_AI5 /initialize get exec
Adobe_IllustratorA_AI5 /initialize get exec
[
39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
/guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe
/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
/hungarumlaut/ogonek/caron
TE
%AI3_BeginEncoding: _Symbol Symbol
[/_Symbol/Symbol 0 0 0 TZ
%AI3_EndEncoding TrueType
%AI3_BeginEncoding: _Times-Roman Times-Roman
[/_Times-Roman/Times-Roman 0 0 1 TZ
%AI3_EndEncoding TrueType
%AI5_Begin_NonPrinting
Np
%AI3_BeginPattern: (Bird's Feet)
(Bird's Feet) 17 7.779 89 79.779 [
(0 O 0 R 1 g 1 G) @
_ &
(0 O 0 R 0 g 0 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
21.323 72.563 m
15.159 75.88 L
S
14.211 74.119 m
20.376 70.802 L
S
18.25 65.779 m
12.952 61.203 L
S
14.26 59.689 m
19.557 64.265 L
S
22.521 57.891 m
16.002 55.34 L
S
16.731 53.478 m
23.25 56.029 L
S
21.194 45.281 m
14.366 46.823 L
S
13.925 44.872 m
20.754 43.33 L
S
21.19 39.635 m
18.63 33.119 L
S
20.492 32.388 m
23.052 38.903 L
S
21.242 23.717 m
14.458 25.441 L
S
13.965 23.503 m
20.75 21.779 L
S
21.02 17.622 m
14.373 15.428 L
S
15 13.529 m
21.647 15.723 L
S
86.211 74.119 m
92.376 70.802 L
S
93.323 72.563 m
87.159 75.88 L
S
86.26 59.689 m
91.557 64.265 L
S
90.25 65.779 m
84.952 61.203 L
S
88.731 53.478 m
95.25 56.029 L
S
94.521 57.891 m
88.002 55.34 L
S
85.925 44.872 m
92.754 43.33 L
S
93.194 45.281 m
86.366 46.823 L
S
85.965 23.503 m
92.75 21.779 L
S
93.242 23.717 m
86.458 25.441 L
S
87 13.529 m
93.647 15.723 L
S
93.02 17.622 m
86.373 15.428 L
S
80.699 75.084 m
84.179 81.158 L
S
82.443 82.152 m
78.963 76.078 L
S
66 74.279 m
63.624 80.863 L
S
61.742 80.184 m
64.118 73.6 L
S
53.43 75.941 m
55.153 82.726 L
S
53.215 83.218 m
51.491 76.434 L
S
47.785 76.096 m
41.341 78.829 L
S
40.56 76.988 m
47.004 74.255 L
S
31.872 76.469 m
33.776 83.205 L
S
31.852 83.749 m
29.947 77.013 L
S
25.785 76.854 m
23.769 83.558 L
S
21.853 82.982 m
23.87 76.278 L
S
82.443 10.152 m
78.963 4.078 L
S
80.699 3.084 m
84.179 9.158 L
S
68.018 10.489 m
72.451 5.071 L
S
73.998 6.338 m
69.565 11.755 L
S
61.742 8.184 m
64.118 1.6 L
S
66 2.279 m
63.624 8.863 L
S
53.215 11.218 m
51.491 4.434 L
S
53.43 3.941 m
55.153 10.726 L
S
31.852 11.749 m
29.947 5.013 L
S
31.872 4.469 m
33.776 11.205 L
S
21.853 10.982 m
23.87 4.278 L
S
25.785 4.854 m
23.769 11.558 L
S
35.443 32.652 m
31.963 26.578 L
S
33.699 25.584 m
37.179 31.658 L
S
65.498 64.838 m
61.065 70.255 L
S
59.518 68.989 m
63.951 63.571 L
S
53.242 66.684 m
55.618 60.1 L
S
57.5 60.779 m
55.124 67.363 L
S
79.785 18.596 m
73.341 21.329 L
S
72.56 19.488 m
79.004 16.755 L
S
79.98 36.559 m
78.075 29.823 L
S
80 29.279 m
81.904 36.014 L
S
69.568 41.906 m
71.584 35.202 L
S
73.5 35.779 m
71.483 42.482 L
S
50.02 47.205 m
53.5 53.279 L
S
51.764 54.273 m
48.284 48.199 L
S
35.518 53.989 m
39.951 48.571 L
S
41.498 49.838 m
37.065 55.255 L
S
27.742 52.184 m
30.118 45.6 L
S
32 46.279 m
29.624 52.863 L
S
79.715 54.218 m
77.991 47.434 L
S
79.93 46.941 m
81.653 53.726 L
S
63.892 50.586 m
70.336 47.853 L
S
71.117 49.694 m
64.673 52.428 L
S
47.068 20.406 m
49.084 13.702 L
S
51 14.279 m
48.983 20.982 L
S
65.969 19.33 m
59.037 18.355 L
S
59.316 16.375 m
66.248 17.349 L
S
66.112 31.548 m
64.55 24.725 L
S
66.5 24.279 m
68.062 31.102 L
S
47.223 34.478 m
43.478 28.564 L
S
45.168 27.494 m
48.913 33.407 L
S
48.892 43.075 m
42.475 40.277 L
S
43.275 38.443 m
49.691 41.241 L
S
35.372 37.469 m
37.276 44.205 L
S
35.352 44.749 m
33.447 38.013 L
S
60.915 40.855 m
58.899 47.558 L
S
56.983 46.982 m
59 40.279 L
S
47.201 64.02 m
41.612 59.806 L
S
42.816 58.209 m
48.405 62.423 L
S
37.937 63.837 m
35.993 70.561 L
S
34.072 70.006 m
36.015 63.281 L
S
27.396 70.357 m
26.989 63.369 L
S
28.986 63.252 m
29.393 70.241 L
S
40.25 22.502 m
35.996 16.943 L
S
37.584 15.727 m
41.839 21.286 L
S
32.456 18.091 m
27.607 23.139 L
S
26.165 21.754 m
31.014 16.705 L
S
72.5 55.279 m
76.901 60.722 L
S
75.346 61.979 m
70.944 56.536 L
S
75.529 65.445 m
76.313 72.401 L
S
74.326 72.625 m
73.542 65.669 L
S
52.56 30.488 m
59.004 27.755 L
S
59.785 29.596 m
53.341 32.329 L
S
68.018 82.489 m
72.451 77.071 L
S
73.998 78.338 m
69.565 83.755 L
S
7.498 64.838 m
3.065 70.255 L
S
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Blue Dots-Transparent)
(Blue Dots-Transparent) 3.88 3.88 32.68 32.68 [
%AI3_Tile
(0 O 0 R 1 0.7 0 0 k 1 0.7 0 0 K) @
(
800 Ar
0 J 0 j 0.5 w 4 M []0 d
%AI3_Note:
0 D
32.68 1 m
34.27 1 35.56 2.29 35.56 3.88 c
35.56 5.471 34.27 6.76 32.68 6.76 c
31.089 6.76 29.8 5.471 29.8 3.88 c
29.8 2.29 31.089 1 32.68 1 c
f
18.28 1 m
19.87 1 21.16 2.29 21.16 3.88 c
21.16 5.471 19.87 6.76 18.28 6.76 c
16.689 6.76 15.4 5.471 15.4 3.88 c
15.4 2.29 16.689 1 18.28 1 c
f
3.88 1 m
5.47 1 6.76 2.29 6.76 3.88 c
6.76 5.471 5.47 6.76 3.88 6.76 c
2.29 6.76 1 5.471 1 3.88 c
1 2.29 2.29 1 3.88 1 c
f
32.68 15.4 m
34.27 15.4 35.56 16.69 35.56 18.28 c
35.56 19.871 34.27 21.16 32.68 21.16 c
31.089 21.16 29.8 19.871 29.8 18.28 c
29.8 16.69 31.089 15.4 32.68 15.4 c
f
18.28 15.4 m
19.87 15.4 21.16 16.69 21.16 18.28 c
21.16 19.871 19.87 21.16 18.28 21.16 c
16.689 21.16 15.4 19.871 15.4 18.28 c
15.4 16.69 16.689 15.4 18.28 15.4 c
f
3.88 15.4 m
5.47 15.4 6.76 16.69 6.76 18.28 c
6.76 19.871 5.47 21.16 3.88 21.16 c
2.29 21.16 1 19.871 1 18.28 c
1 16.69 2.29 15.4 3.88 15.4 c
f
32.68 29.8 m
34.27 29.8 35.56 31.09 35.56 32.68 c
35.56 34.271 34.27 35.56 32.68 35.56 c
31.089 35.56 29.8 34.271 29.8 32.68 c
29.8 31.09 31.089 29.8 32.68 29.8 c
f
18.28 29.8 m
19.87 29.8 21.16 31.09 21.16 32.68 c
21.16 34.271 19.87 35.56 18.28 35.56 c
16.689 35.56 15.4 34.271 15.4 32.68 c
15.4 31.09 16.689 29.8 18.28 29.8 c
f
3.88 29.8 m
5.47 29.8 6.76 31.09 6.76 32.68 c
6.76 34.271 5.47 35.56 3.88 35.56 c
2.29 35.56 1 34.271 1 32.68 c
1 31.09 2.29 29.8 3.88 29.8 c
f
11.08 8.2 m
12.67 8.2 13.96 9.49 13.96 11.08 c
13.96 12.671 12.67 13.96 11.08 13.96 c
9.489 13.96 8.2 12.671 8.2 11.08 c
8.2 9.49 9.489 8.2 11.08 8.2 c
f
25.48 8.2 m
27.07 8.2 28.36 9.49 28.36 11.08 c
28.36 12.671 27.07 13.96 25.48 13.96 c
23.889 13.96 22.6 12.671 22.6 11.08 c
22.6 9.49 23.889 8.2 25.48 8.2 c
f
11.08 22.6 m
12.67 22.6 13.96 23.89 13.96 25.48 c
13.96 27.071 12.67 28.36 11.08 28.36 c
9.489 28.36 8.2 27.071 8.2 25.48 c
8.2 23.89 9.489 22.6 11.08 22.6 c
f
25.48 22.6 m
27.07 22.6 28.36 23.89 28.36 25.48 c
28.36 27.071 27.07 28.36 25.48 28.36 c
23.889 28.36 22.6 27.071 22.6 25.48 c
22.6 23.89 23.889 22.6 25.48 22.6 c
f
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Bricks)
(Bricks) 2.565 5.19 74.565 77.19 [
(0 O 0 R 0.3 0.85 0.85 0 k 0.3 0.85 0.85 0 K) @
_ &
(0 O 0 R 1 g 1 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
1.6 73.6 m
75.6 73.6 l
S
1.6 66.399 m
75.6 66.399 L
S
1.6 59.199 m
75.6 59.199 L
S
1.6 51.998 m
75.6 51.998 L
S
1.6 44.798 m
75.6 44.798 L
S
1.6 37.597 m
75.6 37.597 L
S
1.6 30.397 m
75.6 30.397 L
S
1.6 23.196 m
75.6 23.196 L
S
1.6 15.996 m
75.6 15.996 L
S
1.6 8.796 m
75.6 8.796 L
S
70.975 73.6 m
70.975 66.412 l
S
56.575 73.6 m
56.575 66.412 L
S
42.175 73.6 m
42.175 66.412 L
S
27.775 73.6 m
27.775 66.412 L
S
13.375 73.6 m
13.375 66.412 L
S
70.975 59.162 m
70.975 51.975 l
S
56.575 59.162 m
56.575 51.975 L
S
42.175 59.162 m
42.175 51.975 L
S
27.775 59.162 m
27.775 51.975 L
S
13.375 59.162 m
13.375 51.975 L
S
70.975 44.787 m
70.975 37.6 l
S
56.575 44.787 m
56.575 37.6 L
S
42.175 44.787 m
42.175 37.6 L
S
27.775 44.787 m
27.775 37.6 L
S
13.375 44.787 m
13.375 37.6 L
S
70.975 30.412 m
70.975 23.225 l
S
56.575 30.412 m
56.575 23.225 L
S
42.175 30.412 m
42.175 23.225 L
S
27.775 30.412 m
27.775 23.225 L
S
13.375 30.412 m
13.375 23.225 L
S
70.975 15.975 m
70.975 8.787 l
S
56.575 15.975 m
56.575 8.787 L
S
42.175 15.975 m
42.175 8.787 L
S
27.775 15.975 m
27.775 8.787 L
S
13.375 15.975 m
13.375 8.787 L
S
63.762 8.787 m
63.762 1.6 L
S
49.362 8.787 m
49.362 1.6 L
S
34.962 8.787 m
34.962 1.6 L
S
20.562 8.787 m
20.562 1.6 L
S
6.162 8.787 m
6.162 1.6 l
S
63.762 23.225 m
63.762 16.037 L
S
49.362 23.225 m
49.362 16.037 L
S
34.962 23.225 m
34.962 16.037 L
S
20.562 23.225 m
20.562 16.037 L
S
6.162 23.225 m
6.162 16.037 l
S
63.762 37.6 m
63.762 30.412 L
S
49.362 37.6 m
49.362 30.412 L
S
20.562 37.6 m
20.562 30.412 L
S
6.162 37.6 m
6.162 30.412 l
S
63.762 51.975 m
63.762 44.787 L
S
49.362 51.975 m
49.362 44.787 L
S
34.962 51.975 m
34.962 44.787 L
S
20.562 51.975 m
20.562 44.787 L
S
6.162 51.975 m
6.162 44.787 l
S
63.762 66.412 m
63.762 59.225 L
S
49.362 66.412 m
49.362 59.225 L
S
34.962 66.412 m
34.962 59.225 L
S
20.562 66.412 m
20.562 59.225 L
S
6.162 66.412 m
6.162 59.225 l
S
63.762 80.849 m
63.762 73.662 L
S
49.362 80.849 m
49.362 73.662 L
S
34.962 80.849 m
34.962 73.662 L
S
20.562 80.849 m
20.562 73.662 L
S
6.162 80.849 m
6.162 73.662 l
S
34.962 37.6 m
34.962 30.412 L
S
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Confetti)
(Confetti) 4.85 3.617 76.85 75.617 [
(0 O 0 R 1 g 1 G) @
_ &
(0 O 0 R 0 g 0 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
10.6 64.867 m
7.85 62.867 l
S
9.1 8.617 m
6.85 6.867 l
S
78.1 68.617 m
74.85 67.867 l
S
76.85 56.867 m
74.35 55.117 l
S
79.6 51.617 m
76.6 51.617 l
S
76.35 44.117 m
73.6 45.867 l
S
78.6 35.867 m
76.6 34.367 l
S
76.1 23.867 m
73.35 26.117 l
S
78.1 12.867 m
73.85 13.617 l
S
68.35 14.617 m
66.1 12.867 l
S
76.6 30.617 m
73.6 30.617 l
S
62.85 58.117 m
60.956 60.941 l
S
32.85 59.617 m
31.196 62.181 l
S
47.891 64.061 m
49.744 66.742 l
S
72.814 2.769 m
73.928 5.729 l
S
67.976 2.633 m
67.35 5.909 l
S
61.85 27.617 m
59.956 30.441 l
S
53.504 56.053 m
51.85 58.617 l
S
52.762 1.779 m
52.876 4.776 l
S
45.391 5.311 m
47.244 7.992 l
S
37.062 3.375 m
35.639 5.43 l
S
55.165 34.828 m
57.518 37.491 l
S
20.795 3.242 m
22.12 5.193 l
S
14.097 4.747 m
15.008 8.965 l
S
9.736 1.91 m
8.073 4.225 l
S
31.891 5.573 m
32.005 8.571 l
S
12.1 70.367 m
15.6 68.867 l
S
9.35 54.867 m
9.6 58.117 l
S
12.85 31.867 m
14.35 28.117 l
S
10.1 37.367 m
12.35 41.117 l
S
34.1 71.117 m
31.85 68.617 l
S
38.35 71.117 m
41.6 68.367 l
S
55.1 71.117 m
58.35 69.117 l
S
57.35 65.117 m
55.35 61.867 l
S
64.35 66.367 m
69.35 68.617 l
S
71.85 62.867 m
69.35 61.117 l
S
23.6 70.867 m
23.6 67.867 l
S
20.6 65.867 m
17.35 65.367 l
S
24.85 61.367 m
25.35 58.117 l
S
25.85 65.867 m
29.35 66.617 l
S
14.1 54.117 m
16.85 56.117 l
S
12.35 11.617 m
12.6 15.617 l
S
12.1 19.867 m
14.35 22.367 l
S
26.1 9.867 m
23.6 13.367 l
S
34.6 47.117 m
32.1 45.367 l
S
62.6 41.867 m
59.85 43.367 l
S
31.6 35.617 m
27.85 36.367 l
S
36.35 26.117 m
34.35 24.617 l
S
33.85 14.117 m
31.1 16.367 l
S
37.1 9.867 m
35.1 11.117 l
S
34.35 20.867 m
31.35 20.867 l
S
44.6 56.617 m
42.1 54.867 l
S
47.35 51.367 m
44.35 51.367 l
S
44.1 43.867 m
41.35 45.617 l
S
43.35 33.117 m
42.6 30.617 l
S
43.85 23.617 m
41.1 25.867 l
S
44.35 15.617 m
42.35 16.867 l
S
67.823 31.1 m
64.823 31.1 l
S
27.1 32.617 m
29.6 30.867 l
S
31.85 55.117 m
34.85 55.117 l
S
19.6 40.867 m
22.1 39.117 l
S
16.85 35.617 m
19.85 35.617 l
S
20.1 28.117 m
22.85 29.867 l
S
52.1 42.617 m
54.484 44.178 l
S
52.437 50.146 m
54.821 48.325 l
S
59.572 54.133 m
59.35 51.117 l
S
50.185 10.055 m
53.234 9.928 l
S
51.187 15.896 m
53.571 14.075 l
S
58.322 19.883 m
59.445 16.823 l
S
53.1 32.117 m
50.6 30.367 l
S
52.85 24.617 m
49.6 25.617 l
S
61.85 9.117 m
59.1 10.867 l
S
69.35 34.617 m
66.6 36.367 l
S
67.1 23.617 m
65.1 22.117 l
S
24.435 46.055 m
27.484 45.928 l
S
25.437 51.896 m
27.821 50.075 l
S
62.6 47.117 m
65.321 46.575 l
S
19.85 19.867 m
20.35 16.617 l
S
21.85 21.867 m
25.35 22.617 l
S
37.6 62.867 m
41.6 62.117 l
S
38.323 42.1 m
38.823 38.6 l
S
69.35 52.617 m
66.85 53.867 l
S
14.85 62.117 m
18.1 59.367 l
S
9.6 46.117 m
7.1 44.367 l
S
20.6 51.617 m
18.6 50.117 l
S
46.141 70.811 m
47.994 73.492 l
S
69.391 40.561 m
71.244 43.242 l
S
38.641 49.311 m
39.35 52.117 l
S
25.141 16.811 m
25.85 19.617 l
S
36.6 32.867 m
34.6 31.367 l
S
6.1 68.617 m
2.85 67.867 l
S
4.85 56.867 m
2.35 55.117 l
S
7.6 51.617 m
4.6 51.617 l
S
6.6 35.867 m
4.6 34.367 l
S
6.1 12.867 m
1.85 13.617 l
S
4.6 30.617 m
1.6 30.617 l
S
72.814 74.769 m
73.928 77.729 l
S
67.976 74.633 m
67.35 77.909 l
S
52.762 73.779 m
52.876 76.776 l
S
37.062 75.375 m
35.639 77.43 l
S
20.795 75.242 m
22.12 77.193 l
S
9.736 73.91 m
8.073 76.225 l
S
10.1 23.617 m
6.35 24.367 l
S
73.217 18.276 m
71.323 21.1 l
S
28.823 39.6 m
29.505 42.389 l
S
49.6 38.617 m
47.6 37.117 l
S
60.323 73.6 m
62.323 76.6 l
S
60.323 1.6 m
62.323 4.6 l
S
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Cross Texture)
(Cross Texture) 1 1 58.6 58.6 [
(0 O 0 R 1 1 0.2 0 k 1 1 0.2 0 K) @
_ &
(0 O 0 R 1 g 1 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
53.5 55 m
56.5 55 l
S
39.1 55 m
42.1 55 l
S
24.7 55 m
27.7 55 l
S
10.3 55 m
13.3 55 l
S
46.3 47.8 m
49.3 47.8 l
S
31.9 47.8 m
34.9 47.8 l
S
17.5 47.8 m
20.5 47.8 l
S
3.1 47.8 m
6.1 47.8 l
S
53.5 40.6 m
56.5 40.6 l
S
39.1 40.6 m
42.1 40.6 l
S
24.7 40.6 m
27.7 40.6 l
S
10.3 40.6 m
13.3 40.6 l
S
46.3 33.4 m
49.3 33.4 l
S
31.9 33.4 m
34.9 33.4 l
S
17.5 33.4 m
20.5 33.4 l
S
3.1 33.4 m
6.1 33.4 l
S
53.5 26.2 m
56.5 26.2 l
S
39.1 26.2 m
42.1 26.2 l
S
24.7 26.2 m
27.7 26.2 l
S
10.3 26.2 m
13.3 26.2 l
S
46.3 19 m
49.3 19 l
S
31.9 19 m
34.9 19 l
S
17.5 19 m
20.5 19 l
S
3.1 19 m
6.1 19 l
S
53.5 11.8 m
56.5 11.8 l
S
39.1 11.8 m
42.1 11.8 l
S
24.7 11.8 m
27.7 11.8 l
S
10.3 11.8 m
13.3 11.8 l
S
46.3 4.6 m
49.3 4.6 l
S
31.9 4.6 m
34.9 4.6 l
S
17.5 4.6 m
20.5 4.6 l
S
3.1 4.6 m
6.1 4.6 l
S
55 56.5 m
55 53.5 l
S
40.6 56.5 m
40.6 53.5 l
S
26.2 56.5 m
26.2 53.5 l
S
11.8 56.5 m
11.8 53.5 l
S
47.8 49.3 m
47.8 46.3 l
S
33.4 49.3 m
33.4 46.3 l
S
19 49.3 m
19 46.3 l
S
4.6 49.3 m
4.6 46.3 l
S
55 42.1 m
55 39.1 l
S
40.6 42.1 m
40.6 39.1 l
S
26.2 42.1 m
26.2 39.1 l
S
11.8 42.1 m
11.8 39.1 l
S
47.8 34.9 m
47.8 31.9 l
S
33.4 34.9 m
33.4 31.9 l
S
19 34.9 m
19 31.9 l
S
4.6 34.9 m
4.6 31.9 l
S
55 27.7 m
55 24.7 l
S
40.6 27.7 m
40.6 24.7 l
S
26.2 27.7 m
26.2 24.7 l
S
11.8 27.7 m
11.8 24.7 l
S
47.8 20.5 m
47.8 17.5 l
S
33.4 20.5 m
33.4 17.5 l
S
19 20.5 m
19 17.5 l
S
4.6 20.5 m
4.6 17.5 l
S
55 13.3 m
55 10.3 l
S
40.6 13.3 m
40.6 10.3 l
S
26.2 13.3 m
26.2 10.3 l
S
11.8 13.3 m
11.8 10.3 l
S
47.8 6.1 m
47.8 3.1 l
S
33.4 6.1 m
33.4 3.1 l
S
19 6.1 m
19 3.1 l
S
4.6 6.1 m
4.6 3.1 l
S
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Parquet Floor)
(Parquet Floor) 3.85 3.85 75.85 75.85 [
(0 O 0 R 0.26 0.497 0.75 0 k 0.26 0.497 0.75 0 K) @
_ &
(0 O 0 R 0 g 0 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
37.6 6.1 m
37.6 10.598 L
19.6 10.598 L
19.6 6.1 L
37.6 6.1 L
s
73.6 6.1 m
73.6 10.598 L
55.6 10.598 L
55.6 6.1 L
73.6 6.1 L
s
19.6 24.1 m
19.6 28.598 L
1.6 28.598 L
1.6 24.1 L
19.6 24.1 L
s
55.6 24.1 m
55.6 28.598 L
37.6 28.598 L
37.6 24.1 L
55.6 24.1 L
s
82.6 24.1 m
82.6 28.598 L
73.6 28.598 L
73.6 24.1 L
82.6 24.1 L
s
37.6 42.1 m
37.6 46.598 L
19.6 46.598 L
19.6 42.1 L
37.6 42.1 L
s
73.6 42.1 m
73.6 46.598 L
55.6 46.598 L
55.6 42.1 L
73.6 42.1 L
s
19.6 60.1 m
19.6 64.598 L
1.6 64.598 L
1.6 60.1 L
19.6 60.1 L
s
55.6 60.1 m
55.6 64.598 L
37.6 64.598 L
37.6 60.1 L
55.6 60.1 L
s
82.6 60.1 m
82.6 64.598 L
73.6 64.598 L
73.6 60.1 L
82.6 60.1 L
s
37.6 15.098 m
37.6 19.598 L
19.6 19.598 L
19.6 15.098 L
37.6 15.098 L
s
73.6 15.098 m
73.6 19.598 L
55.6 19.598 L
55.6 15.098 L
73.6 15.098 L
s
19.6 33.098 m
19.6 37.598 L
1.6 37.598 L
1.6 33.098 L
19.6 33.098 L
s
55.6 33.098 m
55.6 37.598 L
37.6 37.598 L
37.6 33.098 L
55.6 33.098 L
s
82.6 33.098 m
82.6 37.598 L
73.6 37.598 L
73.6 33.098 L
82.6 33.098 L
s
37.6 51.098 m
37.6 55.598 L
19.6 55.598 L
19.6 51.098 L
37.6 51.098 L
s
73.6 51.098 m
73.6 55.598 L
55.6 55.598 L
55.6 51.098 L
73.6 51.098 L
s
19.6 69.098 m
19.6 73.598 L
1.6 73.598 L
1.6 69.098 L
19.6 69.098 L
s
55.6 69.098 m
55.6 73.598 L
37.6 73.598 L
37.6 69.098 L
55.6 69.098 L
s
82.6 69.098 m
82.6 73.598 L
73.6 73.598 L
73.6 69.098 L
82.6 69.098 L
s
15.1 19.598 m
10.6 19.598 L
10.6 1.6 L
15.1 1.6 L
15.1 19.598 L
s
51.1 19.598 m
46.6 19.598 L
46.6 1.6 L
51.1 1.6 L
51.1 19.598 L
s
33.1 37.598 m
28.6 37.598 L
28.6 19.6 L
33.1 19.6 L
33.1 37.598 L
s
69.1 37.598 m
64.6 37.598 L
64.6 19.6 L
69.1 19.6 L
69.1 37.598 L
s
15.1 55.598 m
10.6 55.598 L
10.6 37.6 L
15.1 37.6 L
15.1 55.598 L
s
51.1 55.598 m
46.6 55.598 L
46.6 37.6 L
51.1 37.6 L
51.1 55.598 L
s
33.1 73.598 m
28.6 73.598 L
28.6 55.6 L
33.1 55.6 L
33.1 73.598 L
s
69.1 73.598 m
64.6 73.598 L
64.6 55.6 L
69.1 55.6 L
69.1 73.598 L
s
15.1 82.598 m
10.6 82.598 L
10.6 73.6 L
15.1 73.6 L
15.1 82.598 L
s
51.1 82.598 m
46.6 82.598 L
46.6 73.6 L
51.1 73.6 L
51.1 82.598 L
s
19.6 19.598 m
15.1 19.598 L
15.1 1.6 L
19.6 1.6 L
19.6 19.598 L
s
55.6 19.598 m
51.1 19.598 L
51.1 1.6 L
55.6 1.6 L
55.6 19.598 L
s
37.6 37.598 m
33.1 37.598 L
33.1 19.6 L
37.6 19.6 L
37.6 37.598 L
s
73.6 37.598 m
69.1 37.598 L
69.1 19.6 L
73.6 19.6 L
73.6 37.598 L
s
19.6 55.598 m
15.1 55.598 L
15.1 37.6 L
19.6 37.6 L
19.6 55.598 L
s
55.6 55.598 m
51.1 55.598 L
51.1 37.6 L
55.6 37.6 L
55.6 55.598 L
s
37.6 73.598 m
33.1 73.598 L
33.1 55.6 L
37.6 55.6 L
37.6 73.598 L
s
73.6 73.598 m
69.1 73.598 L
69.1 55.6 L
73.6 55.6 L
73.6 73.598 L
s
19.6 82.598 m
15.1 82.598 L
15.1 73.6 L
19.6 73.6 L
19.6 82.598 L
s
55.6 82.598 m
51.1 82.598 L
51.1 73.6 L
55.6 73.6 L
55.6 82.598 L
s
6.1 19.598 m
1.6 19.598 L
1.6 1.6 L
6.1 1.6 L
6.1 19.598 L
s
42.1 19.598 m
37.6 19.598 L
37.6 1.6 L
42.1 1.6 L
42.1 19.598 L
s
78.1 19.598 m
73.6 19.598 L
73.6 1.6 L
78.1 1.6 L
78.1 19.598 L
s
24.1 37.598 m
19.6 37.598 L
19.6 19.6 L
24.1 19.6 L
24.1 37.598 L
s
60.1 37.598 m
55.6 37.598 L
55.6 19.6 L
60.1 19.6 L
60.1 37.598 L
s
6.1 55.598 m
1.6 55.598 L
1.6 37.6 L
6.1 37.6 L
6.1 55.598 L
s
42.1 55.598 m
37.6 55.598 L
37.6 37.6 L
42.1 37.6 L
42.1 55.598 L
s
78.1 55.598 m
73.6 55.598 L
73.6 37.6 L
78.1 37.6 L
78.1 55.598 L
s
24.1 73.598 m
19.6 73.598 L
19.6 55.6 L
24.1 55.6 L
24.1 73.598 L
s
60.1 73.598 m
55.6 73.598 L
55.6 55.6 L
60.1 55.6 L
60.1 73.598 L
s
6.1 82.598 m
1.6 82.598 L
1.6 73.6 L
6.1 73.6 L
6.1 82.598 L
s
42.1 82.598 m
37.6 82.598 L
37.6 73.6 L
42.1 73.6 L
42.1 82.598 L
s
78.1 82.598 m
73.6 82.598 L
73.6 73.6 L
78.1 73.6 L
78.1 82.598 L
s
37.6 1.6 m
37.6 6.098 L
19.6 6.098 L
19.6 1.6 L
37.6 1.6 L
s
73.6 1.6 m
73.6 6.098 L
55.6 6.098 L
55.6 1.6 L
73.6 1.6 L
s
19.6 19.6 m
19.6 24.098 L
1.6 24.098 L
1.6 19.6 L
19.6 19.6 L
s
55.6 19.6 m
55.6 24.098 L
37.6 24.098 L
37.6 19.6 L
55.6 19.6 L
s
82.6 19.6 m
82.6 24.098 L
73.6 24.098 L
73.6 19.6 L
82.6 19.6 L
s
37.6 37.6 m
37.6 42.098 L
19.6 42.098 L
19.6 37.6 L
37.6 37.6 L
s
73.6 37.6 m
73.6 42.098 L
55.6 42.098 L
55.6 37.6 L
73.6 37.6 L
s
19.6 55.6 m
19.6 60.098 L
1.6 60.098 L
1.6 55.6 L
19.6 55.6 L
s
55.6 55.6 m
55.6 60.098 L
37.6 60.098 L
37.6 55.6 L
55.6 55.6 L
s
82.6 55.6 m
82.6 60.098 L
73.6 60.098 L
73.6 55.6 L
82.6 55.6 L
s
37.6 73.6 m
37.6 78.098 L
19.6 78.098 L
19.6 73.6 L
37.6 73.6 L
s
73.6 73.6 m
73.6 78.098 L
55.6 78.098 L
55.6 73.6 L
73.6 73.6 L
s
) &
] E
%AI3_EndPattern
%AI3_BeginPattern: (Scales)
(Scales) 1.6 9.3475 48.088 55.8355 [
(0 O 0 R 1 g 1 G) @
_ &
(0 O 0 R 0 g 0 G) @
(
800 Ar
0 J 0 j 0.3 w 4 M []0 d
%AI3_Note:
0 D
17.0956 9.3475 m
12.8162 9.3475 9.3475 5.8787 9.3475 1.6 C
9.3475 5.8787 5.8787 9.3475 1.6 9.3475 C
1.6 13.6262 5.0687 17.095 9.3475 17.095 c
13.6268 17.095 17.0956 13.6262 17.0956 9.3475 C
s
32.5918 9.3475 m
28.3125 9.3475 24.8437 5.8787 24.8437 1.6 C
24.8437 5.8787 21.3743 9.3475 17.0956 9.3475 C
17.0956 13.6262 20.5644 17.095 24.8437 17.095 c
29.1224 17.095 32.5918 13.6262 32.5918 9.3475 C
s
48.088 9.3475 m
43.8087 9.3475 40.3399 5.8787 40.3399 1.6 C
40.3399 5.8787 36.8705 9.3475 32.5918 9.3475 C
32.5918 13.6262 36.0606 17.095 40.3399 17.095 c
44.6186 17.095 48.088 13.6262 48.088 9.3475 C
s
17.0956 40.3393 m
12.8162 40.3393 9.3475 36.8699 9.3475 32.5912 C
9.3475 36.8699 5.8787 40.3393 1.6 40.3393 C
1.6 44.6181 5.0687 48.0874 9.3475 48.0874 c
13.6268 48.0874 17.0956 44.6181 17.0956 40.3393 C
s
17.0956 24.8431 m
12.8162 24.8431 9.3475 21.3743 9.3475 17.095 C
9.3475 21.3743 5.8787 24.8431 1.6 24.8431 C
1.6 29.1218 5.0687 32.5912 9.3475 32.5912 c
13.6268 32.5912 17.0956 29.1218 17.0956 24.8431 C
s
32.5918 24.8431 m
28.3125 24.8431 24.8437 21.3743 24.8437 17.095 C
24.8437 21.3743 21.3743 24.8431 17.0956 24.8431 C
17.0956 29.1218 20.5644 32.5912 24.8437 32.5912 c
29.1224 32.5912 32.5918 29.1218 32.5918 24.8431 C
s
48.088 24.8431 m
43.8087 24.8431 40.3399 21.3743 40.3399 17.095 C
40.3399 21.3743 36.8705 24.8431 32.5918 24.8431 C
32.5918 29.1218 36.0606 32.5912 40.3399 32.5912 c
44.6186 32.5912 48.088 29.1218 48.088 24.8431 C
s
32.5918 40.3393 m
28.3125 40.3393 24.8437 36.8699 24.8437 32.5912 C
24.8437 36.8699 21.3743 40.3393 17.0956 40.3393 C
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0A090908070706050504030302010100
>
0
1 %_Br
[
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1 1 0 0 1 50 100 %_Bs
BD
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0 1 Pb
Pn
Pc
1 g
Pc
0 g
Pc
0 0 0 0 k
Pc
0.75 g
Pc
0.5 g
Pc
0.25 g
Pc
0 g
Pc
Bb
2 (Black & White) -4014 4716 0 0 1 0 0 1 0 0 Bg
0 BB
Pc
0.25 0 0 0 k
Pc
0.5 0 0 0 k
Pc
0.75 0 0 0 k
Pc
1 0 0 0 k
Pc
0.25 0.25 0 0 k
Pc
0.5 0.5 0 0 k
Pc
0.75 0.75 0 0 k
Pc
1 1 0 0 k
Pc
Bb
2 (Red & Yellow) -4014 4716 0 0 1 0 0 1 0 0 Bg
0 BB
Pc
0 0.25 0 0 k
Pc
0 0.5 0 0 k
Pc
0 0.75 0 0 k
Pc
0 1 0 0 k
Pc
0 0.25 0.25 0 k
Pc
0 0.5 0.5 0 k
Pc
0 0.75 0.75 0 k
Pc
0 1 1 0 k
Pc
Bb
0 0 0 0 Bh
2 (Yellow & Blue Radial) -4014 4716 0 0 1 0 0 1 0 0 Bg
0 BB
Pc
0 0 0.25 0 k
Pc
0 0 0.5 0 k
Pc
0 0 0.75 0 k
Pc
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Pc
0.25 0 0.25 0 k
Pc
0.5 0 0.5 0 k
Pc
0.75 0 0.75 0 k
Pc
1 0 1 0 k
Pc
(Yellow Stripe) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p
Pc
0.25 0.125 0 0 k
Pc
0.5 0.25 0 0 k
Pc
0.75 0.375 0 0 k
Pc
1 0.5 0 0 k
Pc
0.125 0.25 0 0 k
Pc
0.25 0.5 0 0 k
Pc
0.375 0.75 0 0 k
Pc
0.5 1 0 0 k
Pc
0 0 0 0 k
Pc
0 0.25 0.125 0 k
Pc
0 0.5 0.25 0 k
Pc
0 0.75 0.375 0 k
Pc
0 1 0.5 0 k
Pc
0 0.125 0.25 0 k
Pc
0 0.25 0.5 0 k
Pc
0 0.375 0.75 0 k
Pc
0 0.5 1 0 k
Pc
0 0 0 0 k
Pc
0.125 0 0.25 0 k
Pc
0.25 0 0.5 0 k
Pc
0.375 0 0.75 0 k
Pc
0.5 0 1 0 k
Pc
0.25 0 0.125 0 k
Pc
0.5 0 0.25 0 k
Pc
0.75 0 0.375 0 k
Pc
1 0 0.5 0 k
Pc
0 0 0 0 k
Pc
0.25 0.125 0.125 0 k
Pc
0.5 0.25 0.25 0 k
Pc
0.75 0.375 0.375 0 k
Pc
1 0.5 0.5 0 k
Pc
0.25 0.25 0.125 0 k
Pc
0.5 0.5 0.25 0 k
Pc
0.75 0.75 0.375 0 k
Pc
1 1 0.5 0 k
Pc
0 0 0 0 k
Pc
0.125 0.25 0.125 0 k
Pc
0.25 0.5 0.25 0 k
Pc
0.375 0.75 0.375 0 k
Pc
0.5 1 0.5 0 k
Pc
0.125 0.25 0.25 0 k
Pc
0.25 0.5 0.5 0 k
Pc
0.375 0.75 0.75 0 k
Pc
0.5 1 1 0 k
Pc
0 0 0 0 k
Pc
0.125 0.125 0.25 0 k
Pc
0.25 0.25 0.5 0 k
Pc
0.375 0.375 0.75 0 k
Pc
0.5 0.5 1 0 k
Pc
0.25 0.125 0.25 0 k
Pc
0.5 0.25 0.5 0 k
Pc
0.75 0.375 0.75 0 k
Pc
1 0.5 1 0 k
Pc
PB
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%AI5_BeginLayer
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(Layer 1) Ln
0 A
1 Ap
0 R
0 G
800 Ar
0 J 0 j 2 w 4 M []0 d
%AI3_Note:
0 D
322.5 348 m
337.5 348 L
542 348 L
542 713 L
64 713 L
64 348 L
322.5 348 L
s
0 Ap
0 O
0.74 g
64.375 348 m
64.375 479 l
92.1875 480 93 477 114 469.5 c
133.6191 462.493 124.4727 467.1447 152 454.3333 c
178.5 442 171.3333 441 189 427.6667 c
214.4156 408.485 213 411 222 394.5 c
224.0602 390.7229 240.1513 371.6532 241.5 370.5 c
253 360.6667 237.6667 373.6667 256.5 358.5 c
267.0217 350.0269 277.4996 348 256 348 c
182.5 348 64.375 348 y
b
5 w
138.25 348 m
149 367 l
154.5417 405.9375 163.5417 411.9375 v
172.5417 417.9375 169.5417 410.4375 183.0417 426.9375 c
196.5417 443.4375 181.5417 441.9375 198.0417 456.9375 c
214.5417 471.9375 207.0417 485.4375 231.0417 482.4375 c
255.0417 479.4375 246.0417 477.9375 264.0417 473.4375 c
282.0417 468.9375 276.5 471.5 296 471 c
312.7707 470.57 297.3333 466.6667 332 476 c
367.0026 485.424 372.0417 485.4375 381.0417 501.9375 c
390.0417 518.4375 389.3333 509.3333 390.0417 531.9375 c
390.8292 557.0698 379.5417 563.4375 363.0417 557.4375 c
346.5417 551.4375 340.5417 546.9375 333.0417 546.9375 c
325.5417 546.9375 322.5417 548.4375 303.0417 558.9375 c
283.5417 569.4375 265.5417 572.4375 258.0417 587.4375 c
250.5417 602.4375 247.9817 598.0635 246.0417 612.9375 c
244.5417 624.4375 235 644 256 647 c
277 650 279.0417 641.4375 294.0417 638.4375 c
309.0417 635.4375 304.5417 635.4375 334.5417 630.9375 c
364.5417 626.4375 371.5 624 403 624 c
434.5 624 445 626 450.0417 626.4375 c
456.3818 626.9877 507.5 645.5 506 656 c
504.5 666.5 468 667 450 664 c
435.9633 661.6606 436.1513 661.2638 425 658 c
411.3333 654 422 656 402.0417 651.9375 c
379.7987 647.41 376.5417 653.4375 375.0417 662.4375 c
373.5417 671.4375 373 683 376 689 c
390.5 712.8333 l
S
2 w
253.875 476.125 m
253.875 476.125 l
258.375 488.125 255.4365 488.3535 269.25 498.4375 c
275.5 503 288.7814 510.3645 286 510 c
310.1667 513.1667 297.3333 512.3333 310.875 509.125 c
330.5697 504.4588 319.2255 507.9142 336 495.3333 c
348 486.3333 352.875 480.625 y
S
314 508 m
327.5 508 323.9303 504.7563 339.5 511.8333 c
361.5 521.8333 360.8333 525.8333 y
S
385 554 m
391 573.5 396.5522 573.344 391 590 c
390.3333 592 395.3323 604.3288 394 598.3333 c
396 607.3333 389.3333 607.6667 y
S
389.5 567.625 m
410.5 579.625 403 584.125 425.5 582.625 c
448 581.125 445 576.625 458.5 584.125 c
472 591.625 476.5 591.625 481 597.625 c
S
292.25 565.25 m
278.75 584.75 280.0713 581.7347 278.75 598.25 c
277.75 610.75 275.699 609.0216 286.25 623.75 c
309.2109 655.8017 298.9281 640.3925 316.5 656.5 c
S
307 444 m
323 446 322 444 337 448 c
353.197 452.3192 358 455.5 372 460.5 c
386 465.5 391.5 471 397 478.5 c
403.2863 487.0722 409.6602 495.8095 412 504.5 c
415.5 517.5 413 514.6667 413 531 c
413 538 411 546 414 552 c
416.2361 556.4721 422 559 428 558 c
434 557 433 558 441 558 c
449 558 448.2199 557.5 455.5 557.5 c
462 557.5 455.5 557.5 472 562 c
485.8133 565.7672 484 569 492 575 c
500 581 503 589 505 593 c
507 597 508 600 506 604 c
503.4701 609.0596 500.2197 611.4286 495 615 c
488.6667 619.3333 489.3333 622 478 621 c
467.6001 620.0823 466.3264 619.3959 459 615 c
451.5 610.5 454.5 611.5 446 609 c
437.3126 606.4448 430 607 425 607 c
420 607 419 608 415 611 c
411 614 412 618 406 622 c
400 626 402 631 393 633 c
384 635 383 638 376 634 c
369 630 366 631 364 623 c
362 615 362 620 362 608 c
362 596 362.4552 592.8208 361 587 c
359.5 581 356.5596 581.0518 352 579 c
345.3333 576 346.6667 576.6667 y
347 577 341 575 v
335 573 328 577 323 579 c
318 581 319 582 313 588 c
307 594 306.4926 596.9263 307 602 c
307.5 607 306.5 605 309 612 c
311.4254 618.791 313 617.5 321 625 c
328.8911 632.398 332 635 335 642 c
338 649 343.5 652.5 339 663 c
334.5433 673.3989 330 675 323 677 c
316 679 309.1562 679.2625 306 678 c
301 676 297.6667 677.3333 293 673 c
288.3078 668.643 285 663 y
281.5625 655.4375 279.6875 654.125 v
277.8125 652.8125 268.6409 644.4467 268 644 c
259.75 638.25 264 640 256 638 c
245.2844 635.3211 249.5 639 242 642 c
234.5144 644.9943 226.5277 641.5277 222 637 c
218 633 220.6048 637.8106 217 630 c
214 623.5 214 622.6667 216 615 c
218.1567 606.7327 217 607 223 596 c
229 585 227 586 234 577 c
241 568 251 562 261 555 c
271 548 275 544 276 541 c
277 538 276 536 273 531 c
270 526 269.9225 527.6638 263 522 c
257.5 517.5 251.5 513 246 511 c
240 510 233.5 510.5 v
226.4498 511.0423 222.1058 514.8424 210 510 c
200 506 192 501 187 495 c
182 489 179 487 175 480 c
171 473 164 466.5 161 459.5 c
158 452.5 159 453 157 449 c
156.2692 447.5384 151.5641 441.4615 153 435 c
155 426 155 427.5 156 424 c
157.1656 419.9206 162.3583 416.7304 167 414 c
172.6667 410.6667 178 409 184 408 c
190 407 197.328 408.0289 203 411 c
210 414.6667 208 415.3333 213 421 c
217.771 426.4072 218 433 220 436 c
222 439 229 448 232 449 c
235 450 241 451 246 450 c
251 449 252 450 264 448 c
276 446 268 446 278 445 c
287.9504 444.0049 278.3985 444 292 444 c
303 444 307 444 y
s
424 378 m
S
1 Ap
0 O
0 g
193 448 m
195.2091 448 197 449.7909 197 452 c
197 454.2091 195.2091 456 193 456 c
190.7909 456 189 454.2091 189 452 c
189 449.7909 190.7909 448 193 448 c
b
0 Ap
460 647 m
S
350 713 m
541 605 l
S
1 Ap
0 O
0 g
244.125 621.5 m
246.3341 621.5 248.125 623.2909 248.125 625.5 c
248.125 627.7091 246.3341 629.5 244.125 629.5 c
241.9159 629.5 240.125 627.7091 240.125 625.5 c
240.125 623.2909 241.9159 621.5 244.125 621.5 c
b
247 605.5 m
249.2091 605.5 251 607.2909 251 609.5 c
251 611.7091 249.2091 613.5 247 613.5 c
244.7909 613.5 243 611.7091 243 609.5 c
243 607.2909 244.7909 605.5 247 605.5 c
b
392.1617 600.8383 m
394.3708 600.8383 396.1617 602.6292 396.1617 604.8383 c
396.1617 607.0474 394.3708 608.8383 392.1617 608.8383 c
389.9526 608.8383 388.1617 607.0474 388.1617 604.8383 c
388.1617 602.6292 389.9526 600.8383 392.1617 600.8383 c
b
0 To
1 0 0 1 400.3333 597.3333 0 Tp
TP
0 Tr
1 w
/_Times-Roman 18 Tf
0 Ts
100 Tz
0 Tt
0 TA
%_ 0 XL
36 0 Xb
XB
0 0 5 TC
100 100 200 TW
0 0 0 Ti
0 Ta
0 0 2 2 3 Th
0 Tq
0 0 Tl
0 Tc
0 Tw
(w) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 223 616 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
0 Ts
(v) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 230.5 595 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
0 Ts
(v) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 241.3333 587.3333 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
0 Ts
(\r) TX 
TO
0 To
1 0 0 1 178.6667 466 0 Tp
TP
0 Tr
(u) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 119.6667 415.3333 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
0 Ts
(D) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 427 451 0 Tp
TP
0 Tr
/_Symbol 18 Tf
0 Ts
(\266) Tx 
/_Times-Roman 18 Tf
(R) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
(\r) TX 
TO
0 To
1 0 0 1 418.6667 636.3333 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
0 Ts
(u) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i+1) Tx 
(\r) TX 
TO
0 To
1 0 0 1 475.3333 676.6667 0 Tp
TP
0 Tr
/_Symbol 18 Tf
0 Ts
(g) Tx 
(\r) TX 
TO
0 R
0 G
5 w
417.6667 621.3333 m
419.3235 621.3333 420.6667 622.6765 420.6667 624.3333 c
420.6667 625.9902 419.3235 627.3333 417.6667 627.3333 c
416.0098 627.3333 414.6667 625.9902 414.6667 624.3333 c
414.6667 622.6765 416.0098 621.3333 417.6667 621.3333 c
s
0 To
1 0 0 1 472 538 0 Tp
TP
0 Tr
0 O
0 g
1 w
/_Times-Roman 18 Tf
(u) Tx 
/_Times-Roman 14 Tf
-5 Ts
(i) Tx 
/_Times-Roman 18 Tf
0 Ts
(+) Tx 
/_Symbol 18 Tf
(\266) Tx 
/_Times-Roman 18 Tf
(G) Tx 
/_Times-Roman 14 Tf
-5 Ts
(x) Tx 
(\r) TX 
TO
u
0 Ap
0 R
0 G
2 w
400.875 473.3125 m
424.875 457.3125 l
B
395.8534 476.6602 m
398.8005 475.9618 402.7298 475.5034 405.6024 475.8758 c
401.5337 472.8733 l
400.3274 467.9633 l
399.5662 470.758 397.6318 474.2087 395.8534 476.6602 c
b
U
u
u
499.9583 554.0208 m
524.7083 604.5833 l
B
527.6741 610.3165 m
527.3721 607.3028 527.438 603.3475 528.1876 600.5494 c
524.6728 604.1847 l
519.6462 604.7303 l
522.3155 605.8548 525.4797 608.2291 527.6741 610.3165 c
b
U
U
0 To
1 0 0 1 133 425 0 Tp
TP
0 Tr
1 w
0 Ts
(c) Tx 
(\r) TX 
TO
0 To
1 0 0 1 232.5 616.5 0 Tp
TP
0 Tr
/_Times-Roman 18 Tf
(') Tx 
(\r) TX 
TO
LB
%AI5_EndLayer--
%%PageTrailer
gsave annotatepage grestore showpage
%%Trailer
Adobe_IllustratorA_AI5 /terminate get exec
Adobe_typography_AI5 /terminate get exec
Adobe_level2_AI5 /terminate get exec
%%EOF

EndOfTheIncludedPostscriptMagicCookie

\closepsdump

% Here is the PostScript for expcor02.eps:
\message{Writing file expcor02.eps}
\psdump{expcor02.eps}%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(TM) 5.5
%%For: (Kenneth Alexander) (USC)
%%Title: (expcor02B.eps)
%%CreationDate: (5/23/2000) (1:42 PM)
%%BoundingBox: 85 260 542 688
%%HiResBoundingBox: 85.7083 260.0977 541.5931 687.9792
%%DocumentProcessColors: Black
%%DocumentFonts: Symbol
%%+ Times-Roman
%%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0
%%+ procset Adobe_typography_AI5 1.0 0
%%+ procset Adobe_IllustratorA_AI5 1.0 0
%AI5_FileFormat 1.2
%AI3_ColorUsage: Black&White
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 30 31 582 761
%AI3_DocumentPreview: Header
%AI5_ArtSize: 612 792
%AI5_RulerUnits: 2
%AI5_ArtFlags: 1 0 0 1 0 0 1 1 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 1
%AI5_OpenToView: -54 780 1 686 435 18 0 1 31 70
%AI5_OpenViewLayers: 7
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_level2_AI5 1.0 0
%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.0 
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 21 dict dup begin
	put
	/packedarray where not
	{
		userdict begin
		/packedarray
		{
			array astore readonly
		} bind def
		/setpacking /pop load def
		/currentpacking false def
	 end
		0
	} if
	pop
	userdict /defaultpacking currentpacking put true setpacking
	/initialize
	{
		Adobe_level2_AI5 begin
	} bind def
	/terminate
	{
		currentdict Adobe_level2_AI5 eq
		{
		 end
		} if
	} bind def
	mark
	/setcustomcolor where not
	{
		/findcmykcustomcolor
		{
			5 packedarray
		} bind def
		/setcustomcolor
		{
			exch aload pop pop
			4
			{
				4 index mul 4 1 roll
			} repeat
			5 -1 roll pop
			setcmykcolor
		}
		def
	} if
	
	/gt38? mark {version cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def
	userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put
	userdict /level2?
	systemdict /languagelevel known dup
	{
		pop systemdict /languagelevel get 2 ge
	} if
	put
	level2? not
	{
		/setcmykcolor where not
		{
			/setcmykcolor
			{
				exch .11 mul add exch .59 mul add exch .3 mul add
				1 exch sub setgray
			} def
		} if
		/currentcmykcolor where not
		{
			/currentcmykcolor
			{
				0 0 0 1 currentgray sub
			} def
		} if
		/setoverprint where not
		{
			/setoverprint /pop load def
		} if
		/selectfont where not
		{
			/selectfont
			{
				exch findfont exch
				dup type /arraytype eq
				{
					makefont
				}
				{
					scalefont
				} ifelse
				setfont
			} bind def
		} if
		/cshow where not
		{
			/cshow
			{
				[
				0 0 5 -1 roll aload pop
				] cvx bind forall
			} bind def
		} if
	} if
	cleartomark
	/anyColor?
	{
		add add add 0 ne
	} bind def
	/testColor
	{
		gsave
		setcmykcolor currentcmykcolor
		grestore
	} bind def
	/testCMYKColorThrough
	{
		testColor anyColor?
	} bind def
	userdict /composite?
	level2?
	{
		gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore
		add add add 4 eq
	}
	{
		1 0 0 0 testCMYKColorThrough
		0 1 0 0 testCMYKColorThrough
		0 0 1 0 testCMYKColorThrough
		0 0 0 1 testCMYKColorThrough
		and and and
	} ifelse
	put
	composite? not
	{
		userdict begin
		gsave
		/cyan? 1 0 0 0 testCMYKColorThrough def
		/magenta? 0 1 0 0 testCMYKColorThrough def
		/yellow? 0 0 1 0 testCMYKColorThrough def
		/black? 0 0 0 1 testCMYKColorThrough def
		grestore
		/isCMYKSep? cyan? magenta? yellow? black? or or or def
		/customColor? isCMYKSep? not def
	 end
	} if
 end defaultpacking setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0 
%%CreationDate:(03/26/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 54 dict dup begin
put
/initialize
{
 begin
 begin
	Adobe_typography_AI5 begin
	Adobe_typography_AI5
	{
		dup xcheck
		{
			bind
		} if
		pop pop
	} forall
 end
 end
 end
	Adobe_typography_AI5 begin
} def
/terminate
{
	currentdict Adobe_typography_AI5 eq
	{
	 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef
	/_pntr 0 ddef
	{
		counttomark -1 roll
		dup type dup /marktype eq
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef
			} ifelse
		} ifelse
	} loop
	_tempEncode
} def
/TE
{
	StandardEncoding 256 array copy modifyEncoding
	/_nativeEncoding exch def
} def
%
/TZ
{
	dup type /arraytype eq
	{
		/_wv exch def
	}
	{
		/_wv 0 def
	} ifelse
	/_useNativeEncoding exch def
	pop pop
	findfont _wv type /arraytype eq
	{
		_wv makeblendedfont
	} if
	dup length 2 add dict
 begin
	mark exch
	{
		1 index /FID ne
		{
			def
		} if
		cleartomark mark
	} forall
	pop
	/FontName exch def
	counttomark 0 eq
	{
		1 _useNativeEncoding eq
		{
			/Encoding _nativeEncoding def
		} if
		cleartomark
	}
	{
		/Encoding load 256 array copy
		modifyEncoding /Encoding exch def
	} ifelse
	FontName currentdict
 end
	definefont pop
} def
/tr
{
	_ax _ay 3 2 roll
} def
/trj
{
	_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
	/Tx
	{
		dup
		currentpoint 3 2 roll
		tr _psf
		newpath moveto
		tr _ctm _pss
	} ddef
	/Tj
	{
		dup
		currentpoint 3 2 roll
		trj _pjsf
		newpath moveto
		trj _ctm _pjss
	} ddef
} def
/a1
{
	/Tx
	{
		dup currentpoint 4 2 roll gsave
		dup currentpoint 3 2 roll
		tr _psf
		newpath moveto
		tr _ctm _pss
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll gsave
		dup currentpoint 3 2 roll
		trj _pjsf
		newpath moveto
		trj _ctm _pjss
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/e0
{
	/Tx
	{
		tr _psf
	} ddef
	/Tj
	{
		trj _pjsf
	} ddef
} def
/e1
{
	/Tx
	{
		dup currentpoint 4 2 roll gsave
		tr _psf
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll gsave
		trj _pjsf
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/i0
{
	/Tx
	{
		tr sp
	} ddef
	/Tj
	{
		trj jsp
	} ddef
} def
/i1
{
	W N
} def
/o0
{
	/Tx
	{
		tr sw rmoveto
	} ddef
	/Tj
	{
		trj swj rmoveto
	} ddef
} def
/r0
{
	/Tx
	{
		tr _ctm _pss
	} ddef
	/Tj
	{
		trj _ctm _pjss
	} ddef
} def
/r1
{
	/Tx
	{
		dup currentpoint 4 2 roll currentpoint gsave newpath moveto
		tr _ctm _pss
		grestore 3 1 roll moveto tr sp
	} ddef
	/Tj
	{
		dup currentpoint 4 2 roll currentpoint gsave newpath moveto
		trj _ctm _pjss
		grestore 3 1 roll moveto tr jsp
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix
	_tDict begin
	/W
	{
	} def
	/h
	{
	} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/Tr
{
	_render 3 le
	{
		currentpoint newpath moveto
	} if
	dup 8 eq
	{
		pop 0
	}
	{
		dup 9 eq
		{
			pop 1
		} if
	} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm
{
	_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
	_tm astore pop iTm 0 0 moveto
} def
/Td
{
	_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq
	{
	}
	{
		_renderEnd _render get dup null ne
		{
			load exec
		}
		{
			pop
		} ifelse
	} ifelse
	/_render -1 ddef
} def
/Ta
{
	pop
} def
/Tf
{
	dup 1000 div /_fScl exch ddef
%
	selectfont
} def
/Tl
{
	pop
	0 exch _leading astore pop
} def
/Tt
{
	pop
} def
/TW
{
	3 npop
} def
/Tw
{
	/_cx exch ddef
} def
/TC
{
	3 npop
} def
/Tc
{
	/_ax exch ddef
} def
/Ts
{
	/_rise exch ddef
	currentpoint
	iTm
	moveto
} def
/Ti
{
	3 npop
} def
/Tz
{
	100 div /_hs exch ddef
	iTm
} def
/TA
{
	pop
} def
/Tq
{
	pop
} def
/Th
{
	pop pop pop pop pop
} def
/TX
{
	pop
} def
/Tk
{
	exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
	2 npop
} def
/T*
{
	_leading aload pop neg Td
} def
/T*-
{
	_leading aload pop Td
} def
/T-
{
	_hyphen Tx
} def
/T+
{
} def
/TR
{
	_ctm currentmatrix pop
	_tm astore pop
	iTm 0 0 moveto
} def
/TS
{
	currentfont 3 1 roll
	/_Symbol_ _fScl 1000 mul selectfont
	
	0 eq
	{
		Tx
	}
	{
		Tj
	} ifelse
	setfont
} def
/Xb
{
	pop pop
} def
/Tb /Xb load def
/Xe
{
	pop pop pop pop
} def
/Te /Xe load def
/XB
{
} def
/TB /XB load def
currentdict readonly pop
end
setpacking
%%EndResource
%%BeginResource: procset Adobe_IllustratorA_AI5 1.1 0
%%Title: (Adobe Illustrator (R) Version 5.0 Abbreviated Prolog)
%%Version: 1.1 
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_IllustratorA_AI5_vars 70 dict dup begin
put
/_lp /none def
/_pf
{
} def
/_ps
{
} def
/_psf
{
} def
/_pss
{
} def
/_pjsf
{
} def
/_pjss
{
} def
/_pola 0 def
/_doClip 0 def
/cf currentflat def
/_tm matrix def
/_renderStart
[
/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0
] def
/_renderEnd
[
null null null null /i1 /i1 /i1 /i1
] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading
[
0 0
] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc
{
} def
/discardSave null def
/buffer 256 string def
/beginString null def
/endString null def
/endStringLength null def
/layerCnt 1 def
/layerCount 1 def
/perCent (%) 0 get def
/perCentSeen? false def
/newBuff null def
/newBuffButFirst null def
/newBuffLast null def
/clipForward? false def
end
userdict /Adobe_IllustratorA_AI5 74 dict dup begin
put
/initialize
{
	Adobe_IllustratorA_AI5 dup begin
	Adobe_IllustratorA_AI5_vars begin
	discardDict
	{
		bind pop pop
	} forall
	dup /nc get begin
	{
		dup xcheck 1 index type /operatortype ne and
		{
			bind
		} if
		pop pop
	} forall
 end
	newpath
} def
/terminate
{
 end
 end
} def
/_
null def
/ddef
{
	Adobe_IllustratorA_AI5_vars 3 1 roll put
} def
/xput
{
	dup load dup length exch maxlength eq
	{
		dup dup load dup
		length 2 mul dict copy def
	} if
	load begin
	def
 end
} def
/npop
{
	{
		pop
	} repeat
} def
/sw
{
	dup length exch stringwidth
	exch 5 -1 roll 3 index mul add
	4 1 roll 3 1 roll mul add
} def
/swj
{
	dup 4 1 roll
	dup length exch stringwidth
	exch 5 -1 roll 3 index mul add
	4 1 roll 3 1 roll mul add
	6 2 roll /_cnt 0 ddef
	{
		1 index eq
		{
			/_cnt _cnt 1 add ddef
		} if
	} forall
	pop
	exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
	4 1 roll
	{
		2 npop
		(0) exch 2 copy 0 exch put pop
		gsave
		false charpath currentpoint
		4 index setmatrix
		stroke
		grestore
		moveto
		2 copy rmoveto
	} exch cshow
	3 npop
} def
/jss
{
	4 1 roll
	{
		2 npop
		(0) exch 2 copy 0 exch put
		gsave
		_sp eq
		{
			exch 6 index 6 index 6 index 5 -1 roll widthshow
			currentpoint
		}
		{
			false charpath currentpoint
			4 index setmatrix stroke
		} ifelse
		grestore
		moveto
		2 copy rmoveto
	} exch cshow
	6 npop
} def
/sp
{
	{
		2 npop (0) exch
		2 copy 0 exch put pop
		false charpath
		2 copy rmoveto
	} exch cshow
	2 npop
} def
/jsp
{
	{
		2 npop
		(0) exch 2 copy 0 exch put
		_sp eq
		{
			exch 5 index 5 index 5 index 5 -1 roll widthshow
		}
		{
			false charpath
		} ifelse
		2 copy rmoveto
	} exch cshow
	5 npop
} def
/pl
{
	transform
	0.25 sub round 0.25 add exch
	0.25 sub round 0.25 add exch
	itransform
} def
/setstrokeadjust where
{
	pop true setstrokeadjust
	/c
	{
		curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y
	{
		2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		lineto
	} def
	/L
	/l load def
	/m
	{
		moveto
	} def
}
{
	/c
	{
		pl curveto
	} def
	/C
	/c load def
	/v
	{
		currentpoint 6 2 roll pl curveto
	} def
	/V
	/v load def
	/y
	{
		pl 2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
		pl lineto
	} def
	/L
	/l load def
	/m
	{
		pl moveto
	} def
} ifelse
/d
{
	setdash
} def
/cf
{
} def
/i
{
	dup 0 eq
	{
		pop cf
	} if
	setflat
} def
/j
{
	setlinejoin
} def
/J
{
	setlinecap
} def
/M
{
	setmiterlimit
} def
/w
{
	setlinewidth
} def
/H
{
} def
/h
{
	closepath
} def
/N
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			clip /_doClip 0 ddef
		} if
		newpath
	}
	{
		/CRender
		{
			N
		} ddef
	} ifelse
} def
/n
{
	N
} def
/F
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _pf grestore clip newpath /_lp /none ddef _fc
			/_doClip 0 ddef
		}
		{
			_pf
		} ifelse
	}
	{
		/CRender
		{
			F
		} ddef
	} ifelse
} def
/f
{
	closepath
	F
} def
/S
{
	_pola 0 eq
	{
		_doClip 1 eq
		{
			gsave _ps grestore clip newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			_ps
		} ifelse
	}
	{
		/CRender
		{
			S
		} ddef
	} ifelse
} def
/s
{
	closepath
	S
} def
/B
{
	_pola 0 eq
	{
		_doClip 1 eq
		gsave F grestore
		{
			gsave S grestore clip newpath /_lp /none ddef _sc
			/_doClip 0 ddef
		}
		{
			S
		} ifelse
	}
	{
		/CRender
		{
			B
		} ddef
	} ifelse
} def
/b
{
	closepath
	B
} def
/W
{
	/_doClip 1 ddef
} def
/*
{
	count 0 ne
	{
		dup type /stringtype eq
		{
			pop
		} if
	} if
	newpath
} def
/u
{
} def
/U
{
} def
/q
{
	_pola 0 eq
	{
		gsave
	} if
} def
/Q
{
	_pola 0 eq
	{
		grestore
	} if
} def
/*u
{
	_pola 1 add /_pola exch ddef
} def
/*U
{
	_pola 1 sub /_pola exch ddef
	_pola 0 eq
	{
		CRender
	} if
} def
/D
{
	pop
} def
/*w
{
} def
/*W
{
} def
/`
{
	/_i save ddef
	clipForward?
	{
		nulldevice
	} if
	6 1 roll 4 npop
	concat pop
	userdict begin
	/showpage
	{
	} def
	0 setgray
	0 setlinecap
	1 setlinewidth
	0 setlinejoin
	10 setmiterlimit
	[] 0 setdash
	/setstrokeadjust where {pop false setstrokeadjust} if
	newpath
	0 setgray
	false setoverprint
} def
/~
{
 end
	_i restore
} def
/O
{
	0 ne
	/_of exch ddef
	/_lp /none ddef
} def
/R
{
	0 ne
	/_os exch ddef
	/_lp /none ddef
} def
/g
{
	/_gf exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_gf setgray
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/G
{
	/_gs exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_gs setgray
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/k
{
	_cf astore pop
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_cf aload pop setcmykcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/K
{
	_cs astore pop
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_cs aload pop setcmykcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/x
{
	/_gf exch ddef
	findcmykcustomcolor
	/_if exch ddef
	/_fc
	{
		_lp /fill ne
		{
			_of setoverprint
			_if _gf 1 exch sub setcustomcolor
			/_lp /fill ddef
		} if
	} ddef
	/_pf
	{
		_fc
		fill
	} ddef
	/_psf
	{
		_fc
		ashow
	} ddef
	/_pjsf
	{
		_fc
		awidthshow
	} ddef
	/_lp /none ddef
} def
/X
{
	/_gs exch ddef
	findcmykcustomcolor
	/_is exch ddef
	/_sc
	{
		_lp /stroke ne
		{
			_os setoverprint
			_is _gs 1 exch sub setcustomcolor
			/_lp /stroke ddef
		} if
	} ddef
	/_ps
	{
		_sc
		stroke
	} ddef
	/_pss
	{
		_sc
		ss
	} ddef
	/_pjss
	{
		_sc
		jss
	} ddef
	/_lp /none ddef
} def
/A
{
	pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/discard
{
	save /discardSave exch store
	discardDict begin
	/endString exch store
	gt38?
	{
		2 add
	} if
	load
	stopped
	pop
 end
	discardSave restore
} bind def
userdict /discardDict 7 dict dup begin
put
/pre38Initialize
{
	/endStringLength endString length store
	/newBuff buffer 0 endStringLength getinterval store
	/newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store
	/newBuffLast newBuff endStringLength 1 sub 1 getinterval store
} def
/shiftBuffer
{
	newBuff 0 newBuffButFirst putinterval
	newBuffLast 0
	currentfile read not
	{
	stop
	} if
	put
} def
0
{
	pre38Initialize
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff endString eq
			{
				cleartomark stop
			} if
			shiftBuffer
		} loop
	}
	{
	stop
	} ifelse
} def
1
{
	pre38Initialize
	/beginString exch store
	mark
	currentfile newBuff readstring exch pop
	{
		{
			newBuff beginString eq
			{
				/layerCount dup load 1 add store
			}
			{
				newBuff endString eq
				{
					/layerCount dup load 1 sub store
					layerCount 0 eq
					{
						cleartomark stop
					} if
				} if
			} ifelse
			shiftBuffer
		} loop
	}
	{
	stop
	} ifelse
} def
2
{
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		endString eq
		{
			cleartomark stop
		} if
	} loop
} def
3
{
	/beginString exch store
	/layerCnt 1 store
	mark
	{
		currentfile buffer readline not
		{
		stop
		} if
		dup beginString eq
		{
			pop /layerCnt dup load 1 add store
		}
		{
			endString eq
			{
				layerCnt 1 eq
				{
					cleartomark stop
				}
				{
					/layerCnt dup load 1 sub store
				} ifelse
			} if
		} ifelse
	} loop
} def
end
userdict /clipRenderOff 15 dict dup begin
put
{
	/n /N /s /S /f /F /b /B
}
{
	{
		_doClip 1 eq
		{
			/_doClip 0 ddef clip
		} if
		newpath
	} def
} forall
/Tr /pop load def
/Bb {} def
/BB /pop load def
/Bg {12 npop} def
/Bm {6 npop} def
/Bc /Bm load def
/Bh {4 npop} def
end
/Lb
{
	4 npop
	6 1 roll
	pop
	4 1 roll
	pop pop pop
	0 eq
	{
		0 eq
		{
			(%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard
		}
		{
			/clipForward? true def
			
			/Tx /pop load def
			/Tj /pop load def
			currentdict end clipRenderOff begin begin
		} ifelse
	}
	{
		0 eq
		{
			save /discardSave exch store
		} if
	} ifelse
} bind def
/LB
{
	discardSave dup null ne
	{
		restore
	}
	{
		pop
		clipForward?
		{
			currentdict
		 end
		 end
		 begin
			
			/clipForward? false ddef
		} if
	} ifelse
} bind def
/Pb
{
	pop pop
	0 (%AI5_EndPalette) discard
} bind def
/Np
{
	0 (%AI5_End_NonPrinting--) discard
} bind def
/Ln /pop load def
/Ap
/pop load def
/Ar
{
	72 exch div
	0 dtransform dup mul exch dup mul add sqrt
	dup 1 lt
	{
		pop 1
	} if
	setflat
} def
/Mb
{
	q
} def
/Md
{
} def
/MB
{
	Q
} def
/nc 3 dict def
nc begin
/setgray
{
	pop
} bind def
/setcmykcolor
{
	4 npop
} bind def
/setcustomcolor
{
	2 npop
} bind def
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\begin{document}

\title[Power-Law Corrections]{Power-Law Corrections 
to Exponential Decay of
Connectivities and Correlations in Lattice Models}
\author{Kenneth S. Alexander}
\address{Department of Mathematics DRB 155\\
University of Southern California\\
Los Angeles, CA  90089-1113}
\email{alexandr@math.usc.edu}
\thanks{Research supported by NSF grant DMS-9504462.}

\keywords{exponential decay, power-law correction, 
Ornstein-Zernike behavior, weak mixing, FK model}
\subjclass{Primary: 60K35; Secondary: 82B20,82B43}
\date{\today}

\begin{abstract}
Consider a translation-invariant bond percolation model on the 
integer lattice which has exponential decay of connectivities, that is,
the probability of a connection $0 \lra x$ by a path of open bonds 
decreases like $e^{-m(\theta)|x|}$ for some positive constant 
$m(\theta)$ which may depend on the direction $\theta = x/|x|$.  In 
two and three dimensions, it is shown that if the model has an 
appropriate mixing property and satisfies a special case of the
FKG property, then there is at most a power-law correction to the
exponential decay---there exist $A$ and $C$ such that 
$e^{-m(\theta)|x|} \geq P(0 \lra x) \geq A|x|^{-C}e^{-m(\theta)|x|}$
for all nonzero $x$.  In four or more dimensions, a similar bound
holds with $|x|^{-C}$ replaced by $e^{-C(\log |x|)^{2}}$.  In 
particular the power-law lower bound holds for the 
Fortuin-Kasteleyn random cluster model in two dimensions whenever
the connectivity decays exponentially, since the mixing property is 
known to hold in that case.  Consequently a similar bound holds 
for correlations in the Potts model at supercritical temperatures.
\end{abstract}
\maketitle

\section{Introduction and Statement of Results}
Many quantities encountered in statistical mechanics decay 
at an approximately exponential rate as a function of distance.  
Typical finite-range spin systems have exponential decay 
of correlations at sufficiently high tempreatures, and many 
standard percolation models, such as the Fortuin-Kastelyn
random cluster model \cite{FK}, are known or believed to
have exponential decay of connectivities for those parameter values 
(other than critical points) at which there is no percolation.  For the
modified correlation function
\[
  \rho(0,x) = \frac{q^{2}}{q-1}\cov(\delta_{[\sigma_{0} = i]},
  \delta_{[\sigma_{x} = i]})
\]
of the (free-boundary) $q$-state Potts model, or for the 
connectivity function
\[
  \rho(0,x) = P(0 \leftrightarrow x)
\]
of a translation-invariant percolation model having the FKG property,
supermultiplicativity holds:
\[
  \rho(0,x+y) \geq \rho(0,x)\rho(0,y),
\]
so $-\log \rho(0,x)$ is a subadditive function of $x$.  (Here $[0 
\leftrightarrow x]$ denotes the event that 0 is connected to $x$ by a 
path of open bonds, and $\sigma_{x}$ denotes the spin at site $x$.)  
 From standard properties of subadditive sequences, this implies that
the limit
\begin{equation} \label{E:sublim}
  m = m(x/|x|) = \lim_{n \to \infty} \frac{1}{n|x|} \log \rho(0,nx)
\end{equation}
exists, and the exponential approximation is an upper bound for the 
actual correlation or connectivity function:
\[
  \rho(0,x) \leq e^{-m|x|}, \quad x \in \Zd .
\]
It is therefore of interest to find lower bounds, establishing results of the form
\begin{equation} \label{E:lowerform}
 \rho(0,x) \geq f(x)e^{-m|x|}, \quad x \in \Zd,
\end{equation}
where $\rho$ is the correlation or connectivity function, $f$ decays 
subexponentially and $| \cdot |$ is the Euclidean norm.

Ornstein and Zernike \cite{OZ} predicted for certain models that the 
analog of the correlation should behave like 
\begin{equation} \label{E:OZbehav}
  |x|^{-(d-1)/2}e^{-m|x|}
\end{equation}
as $|x| \to \infty$, for some constant $m$.  This was verified for a wide 
class of models at very high temperatures by Bricmont and Fr\"{o}hlich
\cite{BF}, for self-avoiding random walk by Chayes and Chayes 
\cite{CC} and Ioffe \cite{Io}, and for Bernoulli percolation at arbitrary 
subcritical densities, with $x$ ``near an axis,'' by Campanino, Chayes
and Chayes \cite{CCC}.  For the two-dimensional Ising model at 
supercritical temperatures, (\ref{E:OZbehav}) can be obtained 
from the exact solution
(see \cite{MW} or Section 7 of \cite{PV}.)  (An exception to 
(\ref{E:OZbehav})
is found for the two-dimensional Ising model at subcritical temperatures 
under ``plus'' or under ``minus'' boundary conditions, where the correct 
exponent on $|x|$ is 2, not 1/2; see \cite{MW}.  The heuristics for this are 
discussed in \cite{BF}.)  In the case of connectivity functions, the 
heuristic for (\ref{E:OZbehav}) in general is as follows; see e.g. \cite{CCC}
or \cite{Io} for more.  For simplicity take $x$ on an axis, and let $H_{x}$
be the hyperplane orthogonal to the axis at $x$.  The sum of $P(0 
\leftrightarrow y)$ over sites $y$ in $H_{x}$
should be approximately $e^{-m|x|}$ with nearly no correction.  Given that
there is a path from 0 to $H_{x}$, it should reach only a few close-together 
sites in $H_{x}$, and from the central limit theorem, since the transverse
fluctuations of different segments of the path are approximately independent,
the location of these sites in $H_{x}$ should be approximately Gaussian 
distributed with variance of order $|x|$.  This Gaussian distribution 
accounts for the factor $|x|^{-(d-1)/2}$.  (Note constants have been 
omitted in this heuristic.)

Thus the form we should seek for the function $f(x)$ in (\ref{E:lowerform})
is an inverse power of $|x|$, that is, a power-law correction to exponential
decay.  We will not attempt to obtain the optimal power $(d-1)/2$.  We will 
instead obtain results of form (\ref{E:lowerform}) with $f(x)$ an inverse power 
of $|x|$ when $d = $ 2 or 3, and with $f(x) = e^{-C(\log |x|)^{2}}$ for some
constant $C$ when $d \geq 4$.  Analogous results for Bernoulli 
percolation at arbitrary subcritical densities are in \cite{Al1} and \cite{Al3}.
Our results have suboptimal powes of $|x|$ for two reasons.  First, we 
wish to work with quite general models and at arbitrary supercritical 
temperatures, which likely makes rigorous proof of precise behavior 
as in (\ref{E:OZbehav}) a particularly difficult problem.  Second, interesting
applications of (\ref{E:lowerform}) do not always require the optimal 
power $(d-1)/2$.  For example, power-law correction results from 
\cite{Al1} are applied in \cite{Al2} to study boundary fluctuations in the
Wulff construction for Bernoulli percolation, and Pfister and Velenik \cite{PV}
use only the existence of a power-law correction (obtained from the exact
solution) for correlations in the two-dimensional Ising model in their study
of the continuum limit of that model.

For simplicity we restrict attention to the integer lattice, but our results
apply to more general lattices.  For $\Lambda \subset \Zd$ let 
$\mathcal{B}(\Lambda)$ denote the set  of all nearest-neighbor
bonds $\langle xy \rangle$ with $x, y \in \Lambda$, and let 
$\overline{\mathcal{B}}(\Lambda)$ denote the set of all nearest-neighbor
lattice bonds $\langle xy \rangle$ with $x$ or $y$ in $\Lambda$.  
A \emph{bond percolation model} on $\Zd$ is a measure $P$ on 
$\{0,1\}^{\mathcal{B}(\Zd)}$.  We consider here only translation-invariant
models.  A \emph{bond configuration} is an element $\omega \in 
\{0,1\}^{\mathcal{B}(\Zd)}$; when convenient we view $\omega$ as a 
subset of $\mathcal{B}(\Zd)$.  A bond $e$ is \emph{open} in a 
configuration $\omega$ if $\omega_{e} = 1$, and \emph{closed}
if $\omega_{e} = 0$.  The configuration $\{\omega_{e}: e \in \mathcal{G}\}$
restricted to a set $\mathcal{G}$ of bonds is denoted $\omega_{\mathcal{G}}$.
$P$ has \emph{positive connection correlations} if 
\[
  P(0 \leftrightarrow x + y) \geq P(0 \leftrightarrow x)P(x \leftrightarrow x + y)
  \quad \text{for all} \ x, y;
\]
this is a special case of the standard FKG property.  We write 
$P_{\Lambda,\rho}$ for $P( \cdot \mid \omega_{\mathcal{B}(\Lambda^{c})}
= \rho_{\mathcal{B}(\Lambda^{c})})$; we assume the latter is given by a 
regular conditional measure.  Let $\mathcal{F}_{\Lambda}$ denote the 
$\sigma$-algebra generated by $\{\omega_{e}: e \in \mathcal{B}(\Lambda)\}$.
$P$ has the \emph{weak mixing property} if for some $C, \lambda > 0$, for all
finite sets $\Delta, \Lambda$ with $\Delta \subset \Lambda$,
\begin{align}
  \sup \{\Var(&P_{\Lambda,\rho}(\omega_{\mathcal{B}(\Delta)} \in \cdot),
    P_{\Lambda,\rho^{\prime}}(\omega_{\mathcal{B}(\Delta)} \in \cdot)): 
    \rho, \rho^{\prime} \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}\} \notag \\
  &\leq C \sum_{x \in \Delta,y \in \Lambda^{c}} e^{-\lambda |x - y|}, \notag
\end{align}
where $\Var(\cdot,\cdot)$ denotes total variation distance between measures.  
Roughly, the influence of the boundary condition on a finite region decays 
exponentially with distance from that region.  Equivalently, for some 
$C, \lambda > 0$, for all sets $\Delta, \Gamma \subset \Zd$,
\begin{align} \label{E:weakmix}
  \sup \{|&P(E \mid F) - P(E)|:  E \in \mathcal{F}_{\Delta}, F \in
    \mathcal{F}_{\Gamma}, P(F) > 0\} \\
  &\leq C \sum_{x \in \Delta,y \in \Gamma} e^{-\lambda |x - y|}. \notag
\end{align}
$P$ has the \emph{ratio weak mixing} property if for some $C, \lambda > 0$,
for all sets $\Delta, \Gamma \subset \Zd$,
\begin{align} \label{E:rweakmix}
  \sup \biggl\{ \biggl|&\frac{P(E \cap F)}{P(E)P(F)} - 1 \biggr|:  
    E \in \mathcal{F}_{\Delta}, 
    F \in \mathcal{F}_{\Gamma}, P(E)P(F) > 0 \biggr\} \\
  &\leq C \sum_{x \in \Delta,y \in \Gamma} e^{-\lambda |x - y|}, \notag
\end{align}
whenever the right side of (\ref{E:rweakmix}) is less than 1.  For $\Lambda 
\subset \Zd$ finite, $\rho \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}$, and 
$\Gamma \subset \Lambda^{c}$ finite, we call $\mathcal{B}(\Gamma)$ a
\emph{controlling region} for $\overline{\mathcal{B}}(\Lambda)$ and $\rho$ 
if for every $\rho^{\prime} \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}$ such that 
$\rho = \rho^{\prime}$ on $\mathcal{B}(\Gamma)$, we have 
$P_{\Lambda,\rho} = P_{\Lambda,\rho^{\prime}}$.  We say $P$ has 
\emph{exponentially bounded controlling regions} if there exist constants
$C, \lambda > 0$ such that for every choice of disjoint finite sets $\Lambda$
and $\Gamma$,
\begin{align} 
  P(&\{\rho \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}: \mathcal{B}(\Gamma)
    \ \text{is not a controlling region for} \ 
    \overline{\mathcal{B}}(\Lambda) \ 
    \text{and} \ \rho \}) \notag \\
  &\leq C \sum_{x \in \Lambda,y \in \Lambda^{c} \backslash \Gamma} 
    e^{-\lambda |x - y|}. \notag
\end{align}

Note that when $P(E)$ is much smaller than the right side of (\ref{E:weakmix}),
the weak mixing condition (\ref{E:weakmix}) allows $P(E \mid F)$ to be many
times larger than $P(E)$, but the ratio weak mixing condition (\ref{E:rweakmix}) 
does not allow this.  Nonetheless, it is proved in \cite{Al4} that if $P$ has 
exponentially bounded controlling regions and the weak mixing property, then
$P$ has the ratio weak mixing property.  We say $P$ has \emph{exponential
decay of connectivities} if there exist $C, \lambda > 0$ such that for all $x$ 
and $y$,
\[
  P(x \leftrightarrow y) \leq Ce^{-\lambda |x - y|}.
\]
Writing $\theta$ for $x/|x|$, when the limit
\[
  \lim_{n \to \infty} \frac{1}{n|x|} \log P(0 \leftrightarrow nx)
\]
exists for all $x \in \Zd$, is finite and depends only on $\theta$ (as 
for example
when $P$ has positive connection correlations), we denote this limit by 
$m(\theta)$ and say $P$ is \emph{nondegenerate}.

Here is the main result of this paper.

\begin{theorem} \label{T:main}
  Suppose 
  \begin{align} \label{E:assump}
    &P \text{ is a nondegenerate translation-invariant bond 
      percolation model on} \\
    &\Zd \text{ which has 
      positive connection correlations, exponential decay of} \notag \\ 
    &\text{connectivities and the ratio weak mixing property.} \notag
  \end{align}

  (i) If $d$ = 2 or 3, then there exist positive finite $A, C$ and $m(\theta)$
  such that
  \begin{equation} \label{E:23bound}
    e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x) \geq \frac{A}{|x|^{C}}
    e^{-m(\theta)|x|} \quad \text{for all nonzero} \ x \in \Zd,
  \end{equation}
  where $\theta = x/|x|$.

  (ii) If $d \geq$ 4, then there exist positive finite $C$ and $m(\theta)$ 
  such that
  \begin{equation} \label{E:4bound}
    e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x) \geq e^{-C(\log |x|)^{2}}
    e^{-m(\theta)|x|} \quad \text{for all nonzero} \ x \in \Zd,
  \end{equation}
  where $\theta = x/|x|$.
\end{theorem}

The proof will be given in Sections 2 and 3.  Theorem \ref{T:main} applies
to site percolation models as well; we restrict attention to bond percolation
to keep the exposition simple.

The only obstacle to proving the superior result (i), instead of (ii), in 
dimension $d \geq$ 4 is the purely geometric Proposition 2.7 of \cite{Al3},
which is proved only for $d$ = 2 and 3; we believe this Proposition 
is true in all dimensions, and certainly we expect that (\ref{E:23bound})
is true in all dimensions.

The Fortuin-Kasteleyn random cluster model (or simply, the 
\emph{FK model}) with parameters $(q,p)$ and free boundary, on a finite 
subgraph $(\Lambda,\mathcal{B}(\Lambda))$ of the lattice $\Zd$, is the
percolation model with probabilities given by the weights
\[
  p^{|\omega|}(1 - p)^{|\mathcal{B}(\Lambda)|-|\omega|}q^{K(\omega)},
  \quad \omega \in \{0,1\}^{\mathcal{B}(\Lambda)},
\]
where $|\omega|$ denotes the number of open bonds in $\omega$ and
$K(\omega)$ denotes the number of connected components in $\omega$.  Here 
$q > 0$ and $p \in [0,1]$.  Taking the limit $\Lambda \nearrow \Zd$ yields 
the FK model, with free boundary, on the full lattice (see \cite{Gr1}.)  This 
model was introduced in \cite{FK}; see also \cite{ACCN} and \cite{Gr1} for
basic properties.  For the $q$-state Potts model at a supercritical temperature 
$T$, for $\beta = 1/T, p = 1 - e^{-\beta}$ and the FK model at $(p,q)$, the
covariance in the Potts model and the connectivity in the FK model are related by
\begin{equation} \label{E:covarconn}
  q^{2}  \cov(\delta_{[\sigma_{0} = i]},\delta_{[\sigma_{x} = i]})
  = (q - 1)P(0 \leftrightarrow x), \quad i = 1,..,q;
\end{equation}
see \cite{ACCN} or \cite{Gr2}.  Thus exponential decay of connectivities 
in the FK model is equivalent to exponential decay of correlations in
the corresponding Potts model.  Further, the critical inverse temperature 
$\beta_{c}(q,d)$ of the Potts model and the percolation critical point
$p_{c}(q,d)$ of the FK model are related by
\[
  p_{c}(q,d) = 1 - e^{-\beta_{c}(q,d)};
\]
again see \cite{ACCN} or \cite{Gr1}.  For $q \geq 1$, the FK model has 
the FKG property \cite{FK} and hence has positive connection correlations.  
For the two-dimensional FK model, the following facts are known.
For $q = 1, q = 2,$ and $q \geq 25.72$, we have $p_{c}(q,2) = 
\tfrac{\sqrt{q}}{1 + \sqrt{q}}$ \cite{LMR}, and the connectivity decays 
exponentially for all $p < p_{c}(q,2)$ \cite{Gr2}.  This is believed to be true
for all $q$; for $2 < q < 25.72$ the connectivity is known to
decay exponentially at least for all $p < \tfrac{\sqrt{q-1}}
{1 + \sqrt{q-1}}$ \cite{Al5}.  For general $q \geq 1$ and $p < p_{c}(q,2)$, 
if the connectivity decays exponentially then the model has the ratio weak 
mixing property \cite{Al4}.  With Theorem \ref{T:main} and 
(\ref{E:covarconn}), these facts yield 
the following results.
\begin{theorem}  \label{T:FKcase}
  Suppose that the FK model on $\mathbb{Z}^{2}$ with parameters $(q,p)$,
  with $q \geq 1$ and $p < p_{c}(q,2)$, has exponential decay of 
  connectivities.  Then there exist positive finite $A, C$ and $m(\theta)$, 
  depending on $p$ and $q$, such that
  \begin{equation}  \label{E:FKbounds}
    e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x) \geq \frac{A}{|x|^C}
    e^{-m(\theta)|x|} \quad \text{for all} \ x \in \mathbb{Z}^{2},
  \end{equation}
  where $\theta = x/|x|$.  In particular (\ref{E:FKbounds}) holds for all
  $p < p_{c}(q,2) = \tfrac{\sqrt{q}}{1 + \sqrt{q}}$ if $q = 2$ or $q \geq 
  25.72$, and (\ref{E:FKbounds}) holds for all $p < \tfrac{\sqrt{q-1}}
  {1 + \sqrt{q-1}}$ if $2 < q < 25.72$.
\end{theorem}

\begin{corollary}  \label{T:Pottscase}
  Suppose that the $q$-state Potts model on $\mathbb{Z}^{2}$ at inverse 
  temperature $\beta < \beta_{c}(q,2)$ has exponential decay of correlations.  
  Then there exist positive finite $A, C$ and $m(\theta)$, depending on
  $\beta$ and $q$, such that
  \begin{equation}
    e^{-m(\theta)|x|} \geq \cov(\delta_{[\sigma_{0} = i]},
    \delta_{[\sigma_{x} = i]}) \geq \frac{A}{|x|^C}
    e^{-m(\theta)|x|} \quad \text{for all} \ x \in \mathbb{Z}^{2}, i \leq q,
  \end{equation}
  where $\theta = x/|x|$.
\end{corollary}

For the FK model in general dimension, exponential decay of connectivities 
implies exponentially bounded controlling regions (see \cite{Al4}), so that
weak mixing and exponential decay of connectivities together imply ratio 
weak mixing.  It is believed that weak mixing and exponential decay of 
connectivities hold whenever $p < p_{c}(q,d)$, in which case Theorem
\ref{T:main} gives a power-law correction for all subcritical $p$, for 
$d = 3$ and $q \geq 1$, and a correction as in (\ref{E:4bound}) for all
subcritical $p$, for $d \geq 4$ and $q \geq 1$.

It is of interest in certain contexts (see e.g. Lemma 4.3 and Theorem 4.1 
of \cite{Al2}) to have an analog of Theorem \ref{T:main}
for connections in halfspaces; this is our next result.

\begin{theorem} \label{T:halfspace}
  Assume (\ref{E:assump}).  Let $H$ be the intersection with $\Zd$ of
  a closed halfspace in $\mathbb{R}^{d}$
  containing 0.

  (i) If $d$ = 2 or 3, then there exist positive finite $A, C$ and $m(\theta)$
  such that
  \begin{equation} \label{E:H23bound}
    e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x \text{ in } 
    \mathcal{B}(H)) \geq \frac{A}{|x|^{C}}
    e^{-m(\theta)|x|} \quad \text{for all nonzero} \ x \in H,
  \end{equation}
  where $\theta = x/|x|$.

  (ii) If $d \geq$ 4, then there exist positive finite $C$ and $m(\theta)$ 
  such that
  \begin{equation} \label{E:H4bound}
    e^{-m(\theta)|x|} \geq P(0 \leftrightarrow x \text{ in } 
    \mathcal{B}(H)) \geq e^{-C(\log |x|)^{2}}
    e^{-m(\theta)|x|} \quad \text{for all nonzero} \ x \in H,
  \end{equation}
  where $\theta = x/|x|$.
\end{theorem}

\section{Proof of Theorem 1.1$(ii)$}
Throughout the paper, $C_{1},C_{2},...$ and $c_{1},c_{2},...$ denote
constants which may depend on the model $P$, but not on $x$.  
Additional parameters on which these constants may depend are listed in
parentheses after the constant, e.g. $C_{9}(C,K)$.  Phrases such as 
``sufficiently large'' or ``small enough'' implicitly mean ``larger/smaller
than a constant depending only on P,'' unless otherwise specified.  
Throughout the paper we
tacitly assume in proofs that $|x|$ and $C$ 
are sufficiently large, in this sense, and assume (\ref{E:assump}).  To
facilitate bookkeeping we will use $C_{i}$ for constants appearing
in statements of results, and $c_{i}$ for constants which appear only 
in the course of proofs.
  
Define
\[
  h(x) = -\log P(0 \leftrightarrow x), \quad x \in \Zd,
\]
so that, by positive connection correlation, $h$ is subadditive:
\[
  h(x + y) \leq h(x) + h(y).
\]
In particular $\{h(nx): n \geq 1\}$ is a subadditive sequence, so by
standard methods the limit
\begin{equation}  \label{E:mdef}
  m(x) = \lim_{n \to \infty} \frac{h(nx)}{n}
\end{equation}
exists, extending the definition (\ref{E:sublim}), and for all $x \in \Zd$,
\begin{equation}  \label{E:mhcompar}
  m(x) \leq h(x).
\end{equation}
In fact for $x \in \mathbb{Q}^{d}$, if we restrict $n$ to those values 
for which $nx \in \Zd$, then the limit in (\ref{E:mdef}) exists, so $m(\cdot)$
extends to $\mathbb{Q}^{d}$.  By exponential decay of correlations, 
$m(x)$ is strictly positive for all $x \neq 0$.  Further, from 
subadditivity, $m(x)$ is finite if and only if $x$ is in the linear
span of $\{e_{i}: P(\omega_{\langle 0e_{i} \rangle} = 1) > 0\}$, where
$e_{i}$ denotes the ith unit coordinate vector.  Under positive connection
correlations, nondegeneracy of $P$ is equivalent to 
\begin{equation}  \label{E:posprob}
  P(\omega_{\langle 0e_{i} \rangle} = 1) > 0 \quad \text{for all} \ i.
\end{equation}
By the arguments in 
\cite{ACC}, $m$ is uniformly continuous and $m$ extends to a function 
on $\mathbb{R}^{d}$ which is continuous, convex, and 
positive-homogeneous of order 1.  In particular,
\[
  m(x) = m(\theta)|x|,
\]
where $\theta = x/|x|$.  Let
\[
  m_{0} = \min_{i} m(e_{i}), \quad M_{0} = \max_{i} m(e_{i}).
\]
It follows from convexity that
\begin{equation}  \label{E:mbounds}
  m_{0}|x|_{\infty} \leq m(x) \leq M_{0}|x|_{1},
\end{equation}
where $|\cdot |_{r}$ denotes the $l^{r}$ norm.  We suppress the $r$ in the
notation for the Euclidean norm, $r = 2$.

Observe that (\ref{E:4bound}) may be rewritten as
\[
  m(x) \leq h(x) \leq m(x) + C(\log |x|)^{2} \quad \text{for all}
  \ x \in \Zd \ \text{with} \ |x| > 1,
\]
which in the terminology of \cite{Al3} is the \emph{general approximation
property}, or \emph{GAP}, with exponent 0 and correction factor 
$(\log |x|)^{2}$, for the subadditive function $h$.  It is proved in \cite{Al3}
that to establish this property, it is sufficient to establish what is called the 
\emph{convex-hull approximation property}, or \emph{CHAP}, with exponent 
0 and correction factor $\log |x|$.  So we give now a description of CHAP.

Let $B_{1} = \{x \in \mathbb{R}^{d}: m(x) \leq 1\}$.  For $x \in 
\mathbb{R}^{d}$ let $T_{x}$ denote a hyperplane tangent to
$\partial (m(x)B_{1})$ at $x$; note that if $\partial B_{1}$ is not smooth, there 
is not necessarily a unique choice of $T_{x}$.  Let $T_{x}^{0}$ denote
the hyperplane through 0 parallel to $T_{x}$.  There is a unique linear 
functional $m_{x}$ on $\mathbb{R}^{d}$ satisfying
\[
  m_{x}(y) = 0 \quad \text{for all} \ y \in T_{x}^{0}, \quad m_{x}(x) = m(x).
\]
The functional $m_{x}$ is a linear approximation to $m$, for vectors
nearly parallel to $x$.  By convexity and symmetry of $m$ we have
\begin{equation}  \label{E:mxbound}
  |m_{x}(y)| \leq m(y) \quad \text{for all} \ y \in \mathbb{R}^{d}.
\end{equation}
For $y \in \mathbb{R}^{d}, m_{x}(y)$ is the $m$-length of a projection
of $y$ onto the line through 0 and $x$.  The value $m_{x}(y)$ may 
therefore be thought of as the amount of progress (measured in the norm
$m$) toward $x$ made by a vector increment of $y$.  Then for fixed $x$,
\[
  s_{x}(y) = h(y) - m_{x}(y)
\]
is a measure of the error or inefficiency associated with an increment of $y$ 
within a path from 0 to $x$.  For $x \in \mathbb{R}^{d}$ and $C > 1$
we define a set of vector increments for which this ``error'' is of order
at most $\log |x|$:
\[
  Q_{x}(C) = \{y \in \Zd: m_{x}(y) \leq m(x), s_{x}(y) \leq C \log |x|\}.
\]
Note that $s_{x}$ is nonnegative and subadditive, by (\ref{E:mhcompar})
and (\ref{E:mxbound}).  For $M > 0$ and $C, t > 1$, we say that $h$ 
satsfies CHAP($M,C,t$) (with exponent 0 and correction factor 
$\log(\cdot)$) if
\[
  \frac{x}{\alpha} \in \Co(Q_{x}(C)) \quad \text{for some} \ \alpha \in
  [1,t], \quad \text{for all} \ x \in \Zd \ \text{with} \ |x| \geq M,
\]
where $\Co(\cdot)$ denotes the convex hull.  Roughly this says that, up 
to a bounded constant, every $x$ is in the convex hull of some sites 
satisfying the desired power-law lower bound, except that $m$ is replaced 
by the linear approximation $m_{x}$.

\begin{remark} \label{cutup}
In \cite{Al3} the definition of $Q_{x}(\cdot)$ requires in addition, for 
some constant $K$, that $|y| \leq K|x|$.  No such requirement is needed 
here because of Lemma 2.4(i) below.

 From (\cite{Al3}, Lemma 1.6), one way to establish CHAP($M,C,t$) is
to find a lattice path $\gamma$ from 0 to $nx$ for some $n$ which can be cut 
up into at most $tn$ increments, each in $Q_{x}(C)$.  That is, there must
exist sites $0 = u_{0}, u_{1},..,u_{k} = nx$ in $\gamma$ such that 
$k \leq tn$ and $u_{i} - u_{i-1} \in Q_{x}(C)$ for all $i \leq k$.  This was
the approach taken in \cite{Al1}, and our approach here is based somewhat 
on the methods employed there.  Loosely the idea is to show that for large $n$, 
the probability that 0 is connected to $nx$ by a path of open bonds which 
\emph{fails} to have this ``cutting-up'' property is strictly less than 
$P(0 \leftrightarrow nx)$.
\end{remark}

By a \emph{path} we always implicitly mean a self-avoiding lattice path, that
is, a sequence $x_{0}, \langle x_{0}x_{1} \rangle, x_{1}, \langle x_{1}x_{2}
\rangle, x_{2},..,x_{n}$ of alternating sites and bonds, with all $x_{i}$
distinct.  An
\emph{open path} is a path in which all bonds are open.  Define
\[
  G_{x} = \{y \in \Zd: m_{x}(y) \leq m(x)\}.
\]
For $D \subset \mathbb{R}^{d}$ and $y \in \mathbb{R}^{d}$ we let $D + y$ 
denote the translate of the set $D$ by the vector $y$.  Let $d(\cdot,\cdot)$ 
denote Euclidean distance, $d(D,E) = \inf \{d(z,w): z \in D, w \in E\}$
and $d(z,D) = d(\{z\},D)$.
For $D \subset \Zd$ let
\[
  \partial D = \{x \in D^{c}: x \ \text{adjacent to} \ D\}, \quad 
  \overline{D} = D \cup \partial D, \quad \partial_{in}D = \partial (D^{c}).
\]
For $D \subset \mathbb{R}^{d}$ and $y \in D \cap \Zd$ let $\Gamma(y,D)$
denote the union of $\{y\}$ and all sites in 
open paths in $\mathcal{B}(D)$ which contain $y$; if $y \notin D$ we 
define $\Gamma(y,D)$ to be empty.  Note that
\begin{equation}  \label{E:gammayd}
  [\Gamma(y,D) = R] \in \mathcal{F}_{\overline{R}} \quad \text{for all}\  
  \ y, D \ \text{and} \ R.
\end{equation}

Given $x$ and $C$, we say a path $\gamma$ is $(x,C)$-\emph{clean} (or 
just \emph{clean} if confusion is unlikely) if for every pair of sites $u, v$ 
in $\gamma$ with $u$ preceding $v$, we have $s_{x}(v-u) < C \log |x|$.
For sites $y, z \in G \subset \Zd$ we say $z$ is 
$(x,C)$-\emph{cleanly reachable from} 
$y$ \emph{inside} $G$ if there exists an 
$(x,C)$-clean path (not necessarily open!) 
from $y$ to $z$ having all sites in $G$.  Note that clean reachability is a 
deterministic property, not dependent on the bond configuration.  If $z$ is 
$(x,C)$-cleanly reachable from $y$ inside $G$, but is adjacent
to some site in $G$ which is not cleanly reachable from $y$ inside $G$,
we say $z$ is \emph{barely} $(x,C)$\emph{-cleanly reachable from} $y$ 
\emph{inside} $G$.
Define
\[
  \tQxC = \{y \in \Zd: y \ \text{is cleanly reachable from 0 inside} \ G_{x}\}
\]
and observe that
\begin{equation}  \label{E:tqq}
  \tQxC \subset Q_{x}(C).
\end{equation}
Finally define
\begin{align}
  \Delta_{x,C}(y,D) = \{z &\in \Gamma \bigl( y,
    (y + \tQxC) \cap D \bigr) \cap \partial_{in}(y + \tQxC): \notag \\
  &\langle zw \rangle \ \text{is open for some} \ w \in (y + G_{x})
    \backslash (y + \tQxC) \}. \notag
\end{align}
Note that every site in  $\Delta_{x,C}(y,D)$ is connected to $y$ by an open
path (not necessarily clean!)
with all sites in $(y + \tQxC) \cap D$, and is barely cleanly 
reachable from
$y$ inside $y + G_{x}$. 

\begin{remark} \label{R:vdbK}
Let $u_{0},..,u_{n}$ be sites of $\Zd$.  For Bernoulli bond percolation, 
from the
FKG-Harris \cite{Ha} and van den Berg-Kesten \cite{vdBK} 
inequalities one has 
\begin{equation} \label{E:prodlbound}
  P(u_{0} \lra u_{1} \lra .. \lra u_{n}) \geq \prod_{i = 1}^{n}
  P(u_{i -1} \lra u_{i})
\end{equation}
and
\begin{equation} \label{E:produbound}
  P(u_{0} \lra u_{1} \lra .. \lra u_{n} \ \text{via disjoint paths}) 
  \leq \prod_{i = 1}^{n}
  P(u_{i -1} \lra u_{i}),
\end{equation}
which together, roughly speaking, allow one to treat distinct segments 
of a path from $u_{0}$ to $u_{n}$ as independent.  Recall that such 
independence underlies the central limit theorem heuristic for 
Ornstein-Zernike behavior as in (\ref{E:OZbehav}).  The near-independence
given by (\ref{E:prodlbound}) and (\ref{E:produbound}) was strongly
exploited in \cite{Al1}, though not in the context of the central limit theorem,
and the lack of an analog of (\ref{E:produbound}) is perhaps the major
difficulty in adapting the methods of \cite{Al1} to other models.  The
ratio weak mixing property substitutes in part for (\ref{E:produbound}),
but its application requires in effect that one specify nonrandom disjoint 
sets of bonds on which the two events of interest are going to occur,
which is not always feasible for pairs of events like $[u_{i-1} \lra u_{i}]$
and $[u_{j-1} \lra u_{j}]$ in the contexts we would like.  Our solution,
again roughly speaking, involves expressing an event $[u \lra v]$ as a union
$\cup_{R} \ [\Gamma(u,D) = R]$ for an appropriate choice of $D$, where the 
union is over an appropriate collection of sets $R$ containing $u$ and 
$v$.  This is helpful because the event $[\Gamma(u,D) = R]$ necessarily 
takes place on the set of bonds $\overline{\mathcal{B}}(R)$ (cf. 
(\ref{E:gammayd}).)
\end{remark}

For $D \subset \Zd$ and $r \geq 0$ we define
\[
 D^{r} = \{x \in \mathbb{R}^{d}: d(x,D) \leq r\}.
\]

\begin{definition} \label{D:skel}
For $y$ and $z$ sites in a path $\gamma$ with $y$ preceding $z$, we let
$\gamma[y,z]$ denote the segment of $\gamma$ from $y$ to $z$.  Suppose
there is a path $\gamma$ of open bonds in $\omega$ from 0 to $z$ for 
some $z$.  For $C > 1, x \in \Zd$ and $r > 0$, we can then define the 
\emph{gapped} $(C,r,x)$-\emph{skeleton} derived from $\gamma$ in 
$\omega$, a finite sequence $\{(u_{i},v_{i},v_{i}^{\prime},w_{i}),
0 \leq i \leq k\}$ of tuples of sites in $\gamma$, iteratively as follows.
Let $u_{0} = 0$ and $D_{0} = \Zd$.  Having defined $u_{0},..,u_{i},
v_{0},..,v_{i-1}, v_{0}^{\prime},..,v_{i-1}^{\prime}, w_{0},..,
w_{i-1}, D_{0},..,D_{i}$, $\Gamma_{0},..,\Gamma_{i-1}$, and $R_{0},
..,R_{i-1}$, let
\begin{align} 
  \Gamma_{i} &= \Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i})
    \notag \\
  R_{i} &= (\Gamma_{i})^{r \log |x|}  \notag \\
  D_{i+1} &= (R_{0} \cup .. \cup R_{i})^{c}.  \notag
\end{align}
Then let $v_{i}^{\prime}$ be the first site of $\gamma[u_{i},z]$ which 
is not in $\Gamma_{i}$, if such $v_{i}^{\prime}$ exists.  If there is no
such $v_{i}^{\prime}$, then $z \in \Gamma_{i}$ and we let 
$v_{i}^{\prime} = v_{i} = w_{i} = z$ and end the construction; 
otherwise let $v_{i}$ be the site in $\gamma$ immediately preceding
$v_{i}^{\prime}$.  Next let $u_{i+1}$ be the first site of $\gamma$ after
$v_{i}$ with the property that $\gamma[u_{i+1},z]$ is contained in
$D_{i+1}$, if such $u_{i + 1}$ exists.  If no such $u_{i+1}$ exists then
$z \in R_{i} \backslash \Gamma_{i}$ and we let
$v_{i}^{\prime} = v_{i} = w_{i} = z$ and end the construction.  
Let $w_{i}$ be the closest site to $u_{i+1}$ in $\Gamma_{i}$.  Note
that $v_{k}^{\prime} = v_{k} = w_{k} = z$.  See Figures 1 and 2.
\end{definition}

\begin{figure}
  \centerline{ \psfig{file=expcor01.eps,width=4in} }
  \caption{A short increment:  $v_{i} \in \Delta_{x,C}(u_{i},D_{i})$.
  The site $v_{i}^{\prime}$ is not cleanly reachable from $u_{i}$.
  The path $\gamma$ is the heavy line; lighter lines represent other
  paths in $\Gamma_{i}$, and boundaries of regions.}
\end{figure}
\begin{figure}
  \centerline{ \psfig{file=expcor02.eps,width=4in} }
  \caption{A long increment:  $v_{i} \in u_{i} + \partial_{in}G_{x}$.
  The path $\gamma$ stays in the cleanly reachable region all the way
  to $u_{i} + \partial G_{x}$.}
\end{figure}
  
 From the definition of $u_{i+1}$, the site $u_{i+1}^{\prime}$ 
immediately preceding $u_{i+1}$ in $\gamma$ must be in 
$\cup_{j=0}^{i} R_{j}$.  Since $\gamma[u_{i},z]$ does not intersect 
$R_{j}$ for $j < i$, we must in fact have $u_{i+1}^{\prime} \in R_{i}$.
Therefore
\begin{equation}  \label{E:uwdist}
  r \log |x| \leq |u_{i+1} - w_{i}| \leq 1 + r \log |x|.
\end{equation}
The gapped $(C,r,x)$-skeleton then has the following properties:
\begin{align}
  &\text{For each} \ i \ \text{there exist both an open path} \ \psi_{i}
    \ \text{from} \ u_{i} \ \text{to} \ w_{i}, \label{E:prop1} \\
  &\text{and the open path}
    \ \gamma[u_{i},v_{i}] \ \text{from} \ u_{i} \ \text{to} \ v_{i}, \ 
    \text{each having all sites in} \ D_{i} \notag \\
  &\text{and all sites
    cleanly reachable from} \ u_{i} \ \text{inside} \ u_{i} + G_{x}.
    \notag \\
  &\text{For} \ i \neq j, \ \text{the clusters} \ \Gamma_{i} \ \text{and} \ 
    \Gamma_{j} \ \text{are separated by a distance} \label{E:prop2} \\
  &\text{of at least} \ r \log |x|. \notag \\
  &\text{For each} \ i \leq k - 1, v_{i} \in (u_{i} + \partial_{in}G_{x})
    \cup \Delta_{x,C}(u_{i},D_{i}). \label{E:prop3}
\end{align}
Note that the paths $\psi_{i}$ are not necessarily segments of the path
$\gamma$, and we need not have $w_{i} \in \gamma$.  For fixed 
$C$, from (\ref{E:prop3}) we divide the indices into two classes,
corresponding to ``short'' and ``long'' increments $v_{i} - u_{i}$,
as follows (see Figures 1 and 2):
\begin{align}
  S\bigl( (u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k} \bigr) &=
    \{i: 0 \leq i \leq k-1, v_{i} \in \Delta_{x,C}(u_{i},D_{i}) 
    \backslash (u_{i} + \partial_{in}G_{x}) \} \notag \\
  L\bigl( (u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k} \bigr) &=
    \{i: 0 \leq i \leq k-1, v_{i} \in u_{i} + \partial_{in}G_{x} \}. \notag 
\end{align}
Set
\[
  p = \min_{i} P(\omega_{\lbrace 0e_{i} \rbrace} = 1),
\]
so $p > 0$ by (\ref{E:posprob}), and note that by positive connection
correlations,
\begin{equation}  \label{E:crudelower}
  P(0 \lra x) \geq p^{|x|_{1}} \quad \text{for all} \ x.
\end{equation}
The next lemma summarizes some basic properties of the quantities
we have defined.

\begin{lemma} \label{L:basicprops}
  (i) Given $C > 1$ there exists a constant $C_{1}(C)$ such that if 
  $y \in Q_{x}(C)$ and $|x| \geq C_{1}(C)$ then
  \[
    m(y) \leq 2m(x) \quad \text{and} \quad |y| \leq 
    2dM_{0}|x|/m_{0}.
  \]

  (ii) For all $y \in \Zd, 0 \leq s_{x}(y) \leq 2|y|_{1}\log \tfrac{1}{p}$.

  (iii) If $y \in \partial_{in}G_{x}$ then $m_{x}(y) 
  \geq m(x) - M_{0}$.
\end{lemma}
\begin{proof}
(i)  Suppose $m(y) > 2m(x)$ and $m_{x}(y) \leq m(x)$.  Then from 
(\ref{E:mhcompar}) and (\ref{E:mxbound}),
\[
  2m(x) < m(y) \leq h(y) = m_{x}(y) + s_{x}(y) \leq m(x) + s_{x}(y),
\]
so from (\ref{E:mbounds}), $s_{x}(y) > m(x) > C \log |x|$,
provided $|x|$ is large 
(depending on $C$.)  Thus $y \notin Q_{x}(C)$ and the first
inequality in (i) follows.  The second inequality then follows from
(\ref{E:mbounds}).

(ii) The fact that $s_{x}$ is nonnegative has already been noted.  From
(\ref{E:mhcompar}), (\ref{E:mxbound}) and  (\ref{E:crudelower}) we have
\[
  s_{x}(y) \leq h(y) + |m_{x}(y)| \leq 2h(y) \leq 2|y|_{1}\log \frac{1}{p}.
\]

(iii) We have $z = y \pm e_{i}$ for some $z \notin G_{x}$ and 
$i \leq d$.  Therefore using (\ref{E:mxbound}) we have $m_{x}(y)
= m_{x}(z) - m_{x}(\pm e_{i}) \geq m(x) - M_{0}$.
\end{proof}

Let $\diam(B)$ denote the $d$-diameter of a set $B$.
The following is immediate from the definition of ratio weak mixing.

\begin{lemma}  \label{L:rweakmix}
Let $P$ be a bond percolation model on $\Zd$ with the ratio weak mixing 
property.  There exists a constant $C_{2}$ as follows.  Suppose $s > 3$ 
and $U, V \subset \Zd$ with $\diam(U) \leq s$ and 
$d(U,V) \geq C_{2}\log s$.  Then for $D \in \mathcal{F}_{U}, 
E \in \mathcal{F}_{V}$ we have $P(D \cap E) \leq 2P(D)P(E)$.
\end{lemma}

For $y \in \Zd$ and $r > 0$, define $B(y,r) = \{z \in \Zd:
|z - y| \leq r\}$.

\begin{lemma}  \label{L:reachball}
Assume (\ref{E:assump}).
There exists $C_{3}$ such that for $r \geq C_{3}$,
\[
  P(0 \lra \partial B(0,r)) \leq e^{-m_{0}r/2d}.
\]
\end{lemma}
\begin{proof}
For $y \in \partial B(0,r)$ we have $m(y) > m_{0}r/d$ by 
(\ref{E:mbounds}), so $P(0 \lra y) \leq e^{-m_{0}r/d}$.  The result
follows easily.
\end{proof}

We say there is an $r$-\emph{near connection} from $y$ to $z$ in the 
configuration $\omega$ if there exist $u, v$ such that $|u - v|
\leq r, y \lra u$ in $\omega$, and $v \lra z$ in $\omega$.

\begin{lemma}  \label{L:rnear}
Assume (\ref{E:assump}).
There exist $C_{4}$ and $C_{5}$ such that if $|y| > 1, x \neq 0$ and 
$r \geq C_{4} \log |y|$ then
\[
  P(\text{there is an r-near connection from 0 to y}) 
  \leq e^{-m_{x}(y)+C_{5}r}.
\] 
\end{lemma}
\begin{proof}
By Lemma \ref{L:reachball}, (\ref{E:mbounds}) and (\ref{E:mxbound})
there exists $c_{1}$ such that
\begin{equation}  \label{E:bigball}
  P(0 \lra \partial B(0,c_{1}|y|)) \leq e^{-m_{x}(y)}.
\end{equation}
Therefore we need only consider $r$-near connections in 
$\mathcal{B}(B(0,c_{1}|y|))$.  Let $E = 
B(0,c_{1}|y|)$ and 
$\Gamma_{0} = \Gamma(0,E \backslash B(y,r))$, 
and for $R \subset \Zd$ let $F(R) = 
(R^{r})^{c} \cap E$, so
\begin{align} \label{E:F(R)}
  &[\Gamma_{0} = R] \in 
    \mathcal{F}_{\overline{R}}, \quad \diam(F(R)) \leq 
    \diam(E) \leq 2c_{1}|y| \\
  &\quad \text{and} \quad d(\overline{R},F(R)) \geq C_{4} \log 
    |y| - 1. \notag 
\end{align}
If there is an $r$-near connection, but not a connection, 
from 0 to $y$ in $\mathcal{B}(E)$ 
in a configuration $\omega$, let $v(\omega)$ be the closest site to
$\Gamma_{0}$ which has an open path
to $y$ in $\mathcal{B}(F(\Gamma_{0}))$, 
and let $u(\omega)$ be the closest site to $v(\omega)$ 
in $\Gamma_{0}$; ties are broken arbitrarily.
The existence of the $r$-near connection implies that $r \leq |u(\omega)
- v(\omega)| \leq r + 1$.  Note that
\[
  m_{x}(v - u) \leq c_{2}r.
\]
Using this, along with (\ref{E:F(R)}) and Lemma \ref{L:rweakmix}, we
obtain
{\allowdisplaybreaks
\begin{align} \label{E:rnearbound}
  P(&\text{there is an} \ r-\text{near connection from 0 to} \ y \ \text{in} \ 
    \mathcal{B}(E)) \\
  &\leq P(0 \lra y) + \sum_{R,u,v} P\bigl(\Gamma_{0}
    = R, u(\omega) = u, v(\omega) = v \bigr) \notag \\
  &\leq P(0 \lra y) + \sum_{R,u,v} P\bigl(\Gamma_{0}
    = R, v \lra y \ \text{in} \ \mathcal{B}(F(R))\bigr) \notag \\
  &\leq P(0 \lra y) + \sum_{R,u,v} 2P\bigl(\Gamma_{0}
    = R)P(v \lra y) \notag \\
  &\leq P(0 \lra y) + \sum_{u,v} 2P(0 \lra u) P(v \lra y) \notag \\
  &\leq e^{-m_{x}(y)} + \sum_{u,v} 2 e^{-m_{x}(u) - m_{x}(y-v)} \notag \\
  &= e^{-m_{x}(y)} + \sum_{u,v} 2 e^{-m_{x}(y) + m_{x}(v - u)} \notag \\
  &\leq e^{-m_{x}(y)} + 2 |E|^{2} e^{-m_{x}(y) + c_{2}r}  \notag \\
  &\leq e^{-m_{x}(y) + c_{3}r},  \notag
\end{align}
}% end of \allow
where the sums are over all $u, v \in E$ with $r \leq |v - u| \leq
r + 1$ and over all possible values $R$ of 
$\Gamma_{0}$ containing $u$.
Together (\ref{E:bigball}) and (\ref{E:rnearbound}) yield the lemma.
\end{proof}

 From (\ref{E:mhcompar}) and (\ref{E:mxbound}), the probability of
an open path $0 \lra y$ is at most $e^{-m_{x}(y)}$.
Consider for some $C$ an $(x,C)$-unclean
open path $0 \lra u \lra v \lra y$ with $s_{x}(v - u) \geq 
C \log |x|$.
One can ask whether
the cost of such a path (measured by the negative log of
the probability) is increased by an amount of order $C \log |x|$, 
meaning 
that the probability is at most $e^{-m_{x}(y) - cC \log |x|}$.  For 
Bernoulli percolation the van den Berg-Kesten inequality
\cite{vdBK} can be used to show there is always such a cost increase.
But for dependent percolation the situation is more complex.  Consider
the situation in which $u$ and $v$ are approximately on the straight line
$[0,y]$, with $v$ closer to 0, so that the path of interest ``doubles
back'' from $u$ to $v$ on the way to $y$.  If the doubling back 
occurs in a narrow enough tube around the straight line $[0,y]$, then 
the three near-parallel segments of the path between approximately
$v$ and $u$ are not far enough apart for (ratio) weak mixing to
ensure that there is any extra cost.  The next lemma, however, shows 
that if after doubling back from $u$ to $v$ (or otherwise traversing an 
expensive segment) the path does not return to a neighborhood
of $u$, then an extra cost is indeed paid.

\begin{lemma} \label{L:uncleancost1} 
Assume (\ref{E:assump}).
There exist $C_{6}$ with the following property:  For every
$a \geq 1$ and $0 < b < C_{6}$, there exist $C_{7}(a,b), C_{8}(a,b)$
and $C_{9}(a,b)$ such that if $C \geq C_{7}, |x| \geq C_{8}$ 
and $|y| \leq a|x|$, then
\begin{align} \label{E:cost}
  P(&\text{for some } u, v \in \Zd \text{ with } s_{x}(v - u) \geq C \log |x|,
    0 \lra y \text{ via a path } \gamma \text{ which} \\
  &\qquad \text{ visits } u \text{ before } v
    \text{ and does not return to } B(u,bC \log |x|)
    \text{ after visiting } v) \notag \\
  &\leq e^{-m_{x}(y) - C_{9}C \log |x|}. \notag
\end{align}
\end{lemma}
\begin{proof}
By Lemma \ref{L:reachball}, (\ref{E:mbounds}) and (\ref{E:mxbound}),
there exists $c_{4}(a)$ such that for $E = B(0,c_{4}(|x| + C\log |x|))$, we 
have
\begin{equation} \label{E:reachE}
  P(0 \lra \partial E) \leq e^{-m_{x}(y) - C \log |x|}\quad \text{for all }
  |y| \leq a|x|,
\end{equation}
so it is sufficient to consider paths $\gamma$ within $\mathcal{B}(E)$.

For $u, v \in \Zd$ let 
\begin{align}
  &B_{u} = B(u,bC \log |x|), \quad
    \tilde{B}_{u} = B(u,\frac{1}{2}bC \log |x|), \notag \\
  &S_{v} = B(v,4dc_{6}C \log |x|), \quad
    \tilde{S}_{v} = B(v,2dc_{6}C \log |x|), \notag
\end{align}
where $0 < b < c_{5}$; here $c_{5} < c_{6}$
are constants to be specified later.  Provided $c_{6}$ is small
enough, we obtain using Lemma \ref{L:basicprops} that for some $c_{7}
< 1/8$,
for all $u, v$ with $s_{x}(v - u) \geq C \log |x|$,
\begin{align} \label{E:BSsep}
  &m(w - t) \geq 2c_{7}C \log |x| \quad \text{and} \quad
    s_{x}(w - t) \geq
    \frac{C}{2} \log |x|  \\
  &\text{for every } t \in \overline{B_{u}}, w \in
    \overline{S_{v}} \notag
\end{align}
so that in particular $B_{u}$ and $S_{v}$ are disjoint.  Further,
again provided $c_{6}$ is small enough, we have
\begin{equation} \label{E:Svar}
  |m_{x}(q - r)| \leq c_{7}C \log |x| \quad \text{for all } q,r \in 
  \overline{S_{0}}
\end{equation}
and if also $c_{5}$ is small enough relative to $c_{6}$,
\begin{equation} \label{E:Bvar}
  |m_{x}(t - s)| < \frac{m_{0}}{4}c_{6}C \log |x|
  < c_{7}C \log |x| \quad \text{for all } s,t \in 
  \overline{B_{0}}.
\end{equation}

Fix $y \in \Zd$.  For $u, v$ with $s_{x}(v - u) \geq C \log |x|$,
let $A(u,v)$ be the event that there exists an open path $\gamma$
from 0 to $y$ in $\mathcal{B}(E)$ which visits $u$ before $v$
and does not return to $B_{u}$ after reaching $v$.  

\emph{Case 1}. $0, y \notin B_{u} \cup S_{v}$.
We can then further 
decompose $A(u,v)$ as follows:  for $s, t \in \partial B_{u}$ and 
$w, z \in \partial S_{v}$, let $A(u,v;s,t,w,z)$ be the event that there
exists $\gamma$ as above which first reaches $\partial B_{u}$ at $s$,
which last exits $B_{u}$ via a step to $t$, which has $\gamma[t,y]$
first enter $S_{v}$ at $w$, and which last exits $S_{v}$ via a step
to $z$.  

Ideally, when $A(u,v;s,t,w,z)$ occurs we would like the 
three segments $\gamma[0,s]$ from 0 to $\partial B_{u}, 
\gamma[t,w]$ from $\partial B_{u}$ to $\partial S_{v}$, and 
$\gamma[z,y]$ from $\partial S_{v}$ to $y$ to be well-separated
from one another, so that Lemma \ref{L:rweakmix} can be applied, 
but in fact there may be various unwanted connections or 
near-connections outside $B_{u}$ and/or $S_{v}$
which we must handle.  Depending on the presence 
of these near-connections, the source of the extra cost
$C_{9}C \log |x|$ exhibited in (\ref{E:cost}) is either the 
expensive segment from $u$ to $v$, or the connection from
$u$ to $\partial \tilde{B}_{u}$ inside $B_{u}$, or the connection
from $v$ to $\partial \tilde{S}_{v}$ inside $S_{v}$.

For $q, r \in A \subset \Zd$, let $N(q,r,A)$ be the 
event that there is a 
$(2 + c_{8}bC \log |x|)$-near
connection from $q$ to $r$ in $\mathcal{B}(A)$, where 
$c_{8}$ is a (small) constant to be specified.  

Note that the distance $2 + c_{8}bC \log |x|$ quantifying
``near-connections'' is much less than the diameter of $B_{u}$;
$B_{u}$, in turn, is much smaller than $S_{v}$.

First consider $A(u,v) \cap N(0,y,E \backslash B_{u})$.  When this
event occurs we have a near-connection from 0 to $y$ outside $B_{u}$,
and a connection from $u$ to $\partial \tilde{B}_{u}$.  If $c_{8}$ is
sufficiently small (depending on $C_{5}$), and $C$ and $x$ are
sufficiently large (depending on $a, b$), then Lemmas 
\ref{L:rweakmix}, \ref{L:reachball}
and \ref{L:rnear} give for some $c_{9}$
\begin{align} \label{E:case1}
  P(&A(u,v) \cap N(0,y,E \backslash B_{u})) \\
  &\leq P([u \lra \partial \tilde{B}_{u}] \cap N(0,y,E \backslash 
    B_{u})) \notag \\
  &\leq 2 P(u \lra \partial \tilde{B}_{u}) P(N(0,y,\Zd)) \notag \\
  &\leq 2e^{-m_{x}(y) + C_{5}c_{8}bC \log |x|
    - m_{0}bC (\log |x|)/4d} \notag \\
  &\leq 2e^{-m_{x}(y) - c_{9}bC \log |x|}. \notag
\end{align}

Second, consider $A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
\cap N(t,y,E \backslash
(B_{u} \cup S_{v}))^{c}$.  When this occurs 
we have clusters $\Gamma(0,E
\backslash B_{u})$ containing the sites of $\gamma[0,s],
\Gamma(t,E \backslash (B_{u} \cup S_{v}))$ containing 
the sites of $\gamma[t,w]$,
and $\Gamma(y,E \backslash (B_{u} \cup S_{v}))$ containing
the sites of $\gamma[z,y]$, these clusters being separated 
from each other by at least $2 + c_{8}bC \log |x|$.  Therefore using 
Lemma \ref{L:rweakmix}, (\ref{E:gammayd}), (\ref{E:BSsep}) 
and (\ref{E:Svar}), if $C$ is
sufficiently large (depending on $b$) and if $|x|$ is sufficiently
large (depending on $a, b$), we obtain that for some $c_{10}$,
{\allowdisplaybreaks
\begin{align} \label{E:case2}
  P(&A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
    \cap N(t,y,E \backslash
    (B_{u} \cup S_{v}))^{c}) \\
  &\leq \sum_{I,J,K} P\bigl(\Gamma(0,E \backslash B_{u}) = I,
    \Gamma(t,E \backslash (B_{u} \cup S_{v})) = J,
    \Gamma(y,E \backslash (B_{u} \cup S_{v})) = K\bigr) \notag \\
  &\leq \sum_{I,J,K} 4 P\bigl(\Gamma(0,E \backslash B_{u}) = I \bigr)
    P\bigl(\Gamma(t,E \backslash (B_{u} \cup S_{v})) = J \bigr)
    P\bigl(\Gamma(y,E \backslash (B_{u} \cup S_{v})) = K \bigr) \notag \\
  &\leq 4 P(0 \lra s) P(t \lra w) P(z \lra y) \notag \\
  &\leq 4 e^{-[m_{x}(s) + m_{x}(w - t) + s_{x}(w - t)
    + m_{x}(y - z)]} \notag \\
  &= 4 e^{-[m_{x}(y) - m_{x}(t - s) - m_{x}(z - w) 
    + s_{x}(w - t)]} \notag \\
  &\leq 4 e^{-m_{x}(y) + 2c_{7}C \log |x| - 
    \frac{C}{4} \log |x|}\notag \\
  &\leq e^{-m_{x}(y) - c_{10}C \log |x|} \notag
\end{align}
}%end of \allow
where the sums are over all $I, J, K \subset E$ with 
$0,s \in I, t, w \in J, z, y \in K$, and $\min(d(I,J)$,
$d(I,K),d(J,K)) > 2 + c_{8}bC \log |x|.$

Third, consider $A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
\cap N(0,s,E \backslash (B_{u} \cup S_{v}))^{c}$.  
When this occurs
and $\Gamma(0,E \backslash (B_{u} \cup S_{v})) = I,
\Gamma(s,E \backslash (B_{u} \cup S_{v})) = J,
\Gamma(0,E \backslash B_{u}) = K$, 
$\Gamma(y,E \backslash (B_{u} \cup S_{v})) = L$
for some $I,J,K,L$, we must have $I \cup J \subset K,
d(I,J) > 2 + c_{8}bC \log |x|$ and
$d(K,L) > 2 + c_{8}bC \log |x|$; $I$ contains 
the sites of an open path from 0 to $\partial S_{v}$,
$J$ contains the sites of an open path from $\partial S_{v}$
to $s$, and $L$ contains the sites of $\gamma[z,y]$.  (Here we
do not make use of the cluster containing $t$ and $w$.) Therefore using 
Lemma \ref{L:rweakmix}, (\ref{E:gammayd}), (\ref{E:BSsep}) 
and (\ref{E:Svar}), if $C$ and $x$ are
sufficiently large (depending on $a, b$), we obtain that for some $c_{11}$,
{\allowdisplaybreaks
\begin{align} \label{E:case3}
  P\bigl(&A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
    \cap N(0,s,E \backslash (B_{u} \cup S_{v}))^{c} \bigr) \\
  &\leq \sum_{I,J,L} P\bigl(\Gamma(0,E \backslash (B_{u} \cup 
    S_{v})) = I, \Gamma(s,E \backslash (B_{u} \cup S_{v})) = J,
    \Gamma(y,E \backslash B_{u}) = L \bigr) \notag \\
  &\leq \sum_{I,J,L} 4P\bigl(\Gamma(0,E \backslash (B_{u} \cup 
    S_{v})) = I \bigr) P\bigl(\Gamma(s,E \backslash (B_{u} 
    \cup S_{v})) = J \bigr)
    P\bigl(\Gamma(y,E \backslash B_{u}) = L \bigr) \notag \\
  &\leq 4 P(0 \lra \partial S_{v}) P(\partial S_{v} \lra s)
    P(z \lra y) \notag \\
  &\leq 4 \sum_{q,r \in \partial S_{v}} P(0 \lra q) P(r \lra s)
    P(z \lra y) \notag \\
  &\leq 4 \sum_{q,r \in \partial S_{v}} e^{-[m_{x}(q) + m(s - r)
    + m_{x}(y - z)]} \notag \\
  &= 4 \sum_{q,r \in \partial S_{v}} e^{-m_{x}(y) + m_{x}(z - q)
    - m(s - r)} \notag \\
  &\leq |\partial S_{v}|^{2} e^{-m_{x}(y) - c_{7}C \log |x|} \notag \\
  &\leq e^{-m_{x}(y) - c_{11}C \log |x|} \notag
\end{align}
}%end of \allow
where the sum is over $I,J,L \subset E$ with $0 \in I, I \cap \partial S_{v} \neq 
\phi, s \in J, J \cap \partial S_{v} \neq \phi, y, z \in L$, and 
$\min(d(I,J),d(I,L),d(J,L)) > 2 + c_{8}bC \log |x|$.

Fourth, consider 
\[
  A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
  \cap N(0,s,E \backslash (B_{u} \cup S_{v})) \cap
  N(t,y,E \backslash (B_{u} \cup S_{v})).
\]
When this occurs
and $\Gamma(0,E \backslash (B_{u} \cup S_{v})) \cup
\Gamma(s,E \backslash (B_{u} \cup S_{v})) = I,
\Gamma(0,E \backslash B_{u}) = J,
\Gamma(t,E \backslash (B_{u} \cup S_{v})) \cup
\Gamma(y,E \backslash (B_{u} \cup S_{v})) = K$ and 
$\Gamma(y,E \backslash B_{u}) = L$ for some
$I, J, K,L$, we must have $I \subset J,
K \subset L$ and $d(J,L) \geq
2 + c_{8}bC \log |x|$; $I$ contains the sites of a near-connection 
from 0 to $s$, and $K$ contains the sites of a near-connection
from $t$ to $y$.  There is also an open path from $v$ to
$\partial \tilde{S}_{v}$.
Therefore assuming $b$ is small enough relative to $c_{6}$,
using 
Lemmas \ref{L:rweakmix} - \ref{L:rnear},
(\ref{E:gammayd}) and (\ref{E:Bvar}),
if $C$ and $x$ are
sufficiently large (depending on $a, b$), we obtain 
{\allowdisplaybreaks
\begin{align} \label{E:case4}
  P\bigl(&A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
    \cap N(0,s,E \backslash (B_{u} \cup S_{v})) \cap
    N(t,y,E \backslash (B_{u} \cup S_{v})) \bigr) \\
  &\leq \sum_{I,K} P\bigl(\Gamma(0,E \backslash (B_{u} \cup S_{v})) 
    \cup \Gamma(s,E \backslash (B_{u} \cup S_{v})) = I, \notag \\
  &\qquad \qquad \Gamma(t,E \backslash (B_{u} \cup S_{v})) \cup
    \Gamma(y,E \backslash (B_{u} \cup S_{v})) = K,
    v \lra \partial \tilde{S}_{v} \bigr) \notag \\
  &\leq \sum_{I,K} 4 P\bigl(\Gamma(0,E \backslash (B_{u} \cup S_{v})) 
    \cup \Gamma(s,E \backslash (B_{u} \cup S_{v})) = I \bigr) \notag \\
  &\qquad \qquad P\bigl(\Gamma(y,E \backslash (B_{u} \cup S_{v})) \cup
    \Gamma(t,E \backslash (B_{u} \cup S_{v})) = K \bigr) 
    P(v \lra \partial \tilde{S}_{v}) \notag \\
  &\leq 4P(N(0,s,\Zd)) P(N(t,y,\Zd)) P(v \lra \partial \tilde{S}_{v}) 
    \notag \\
  &\leq 4e^{-m_{x}(s) - m_{x}(y - t) +2C_{5}c_{8}bC \log |x| 
    - m_{0}c_{6}C \log |x|} \notag \\
  &\leq e^{-m_{x}(y) + m_{x}(t - s) - \frac{1}{2}m_{0}c_{6}C
    \log |x|} \notag \\
  &\leq e^{-m_{x}(y) - \frac{1}{4}m_{0}c_{6}C \log |x|}. \notag
\end{align}
}%end of \allow
where the sum is over those $I \ni 0, s$ and $K \ni t, y$ consistent 
with the event appearing in the first sum in (\ref{E:case4}), with
$d(I,K) \geq 2 + c_{8}bC \log |x|$.  Combining (\ref{E:case1}),
(\ref{E:case2}), (\ref{E:case3}) and (\ref{E:case4})and
summing over $s, t, w, z$, provided $C$ and $x$ are
sufficiently large (depending on $a, b$), we obtain
\begin{equation} \label{E:Auvbound}
  P(A(u,v)) \leq e^{-m_{x}(y)-C_{14}bC \log |x|}.
\end{equation}

It remains to consider cases with 0 and/or $y$ in $B_{u} \cup 
S_{v}$.  Note that when $A(u,v)$ occurs we cannot have $y \in 
B_{u}$.  Also, the bound (\ref{E:case1}) is valid regardless of the
locations of $u$ and $v$ (the left side is 0 if $0 \in B_{u}$.)

\emph{Case 2}.  $0 \in B_{u},\ y \notin B_{u} \cup S_{v}$.  Here
there is no longer a site $s$ but we can define $t, w, z$ and 
$A(u,v;t,w,v)$ similarly to Case 1.  Similarly to (\ref{E:case2}) we
obtain
\begin{align} \label{E:secondcase1}
  P\bigl(&A(u,v,t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
    \cap N(t,y,E \backslash (B_{u} \cup S_{v}))^{c} \bigr) \\
  &\leq 2 P(t \lra w) P(z \lra y) \notag \\
  &\leq 2 e^{-[m_{x}(w - t) + s_{x}(w - t)
    + m_{x}(y - z)]} \notag \\
  &= 2 e^{-[m_{x}(y) - m_{x}(t) - m_{x}(z - w) 
    + s_{x}(w - t)]} \notag \\
  &\leq 2 e^{-m_{x}(y) + 2c_{7}C \log |x| - 
    \frac{C}{4} \log |x|}\notag \\
  &\leq e^{-m_{x}(y) - c_{10}C \log |x|}, \notag
\end{align}
while similarly to (\ref{E:case4}) we obtain
\begin{align} \label{E:secondcase2}
  P\bigl(&A(u,v;t,w,z) \cap N(0,y,E \backslash B_{u})^{c}
    \cap N(t,y,E \backslash (B_{u} \cup S_{v})) \bigr) \\
  &\leq 2 P(N(t,y,\Zd)) P(v \lra \partial \tilde{S}_{v}) 
    \notag \\
  &\leq 4e^{-m_{x}(y - t) + C_{5}c_{8}bC \log |x| 
    - m_{0}c_{6}C \log |x|} \notag \\
  &\leq e^{-m_{x}(y) + m_{x}(t) - \frac{1}{2}m_{0}c_{6}C
    \log |x|} \notag \\
  &\leq e^{-m_{x}(y) - C_{14}C \log |x|}. \notag
\end{align}
Summing these over $t,w,z$ and combining with (\ref{E:case1})
yields (\ref{E:Auvbound}).

\emph{Case 3}.  $0 \in S_{v}, y \notin B_{u} \cup S_{v}$.
Similarly to (\ref{E:case3}), using (\ref{E:BSsep})
and (\ref{E:Svar}) we obtain
\begin{align} \label{E:thirdcase}
  P\bigl(&A(u,v;s,t,w,z) \cap N(0,y,E \backslash B_{u})^{c} \bigr) \\
  &\leq 2 P(0 \lra s) P(z \lra y) \notag \\
  &\leq 2 e^{-[m(s) + m_{x}(y - z)]} \notag \\
  &\leq e^{-\frac{C}{2}\log |x| - m_{x}(y) + c_{7}C \log |x|} \notag \\
  &\leq e^{-m_{x}(y) - \frac{C}{4} \log |x|}. \notag
\end{align}
Summing over $s,t,w,z$ and combining with (\ref{E:case1}) again
yields (\ref{E:Auvbound}).

\emph{Case 4}.  $0 \notin B_{u} \cup S_{v}, y \in S_{v}$.  This 
time
there is no longer a site $z$ but we can define $s,t,w$ and 
$A(u,v;s,t,w)$ similarly to Case 1.  Similarly to (\ref{E:case2}) we
obtain
\begin{align} \label{E:fourthcase}
  P\bigl(&A(u,v;s,t,w) \cap N(0,y,E \backslash B_{u})^{c} \bigr) \\
  &\leq 2 P(0 \lra s) P(t \lra w) \notag \\
  &\leq 2 e^{-[m_{x}(s) + m_{x}(w - t) + s_{x}(w - t)]} \notag \\
  &= 2 e^{-[m_{x}(y) - m_{x}(t-s) - m_{x}(y - w) 
    + s_{x}(w - t)]} \notag \\
  &\leq 2 e^{-m_{x}(y) + 2c_{7}C \log |x| - 
    \frac{C}{4} \log |x|}\notag \\
  &\leq e^{-m_{x}(y) - c_{10}C \log |x|}. \notag
\end{align}
Once again, 
summing over $s,t,w$ and combining with (\ref{E:case1})
yields (\ref{E:Auvbound}).

\emph{Case 5}.  $0 \in B_{u}, y \in S_{v}$.  Here
\begin{align} \label{E:fifthcase}
  P(A(u,v)) &\leq P(u \lra v) \\
  &\leq e^{-[m_{x}(v - u) + s_{x}(v - u)]} \notag \\
  &= e^{-[m_{x}(y) - m_{x}(y - v) - m_{x}(u)
    +s_{x}(v - u)]} \notag \\
  &\leq e^{-m_{x}(y) + 2c_{7}C \log |x| - C \log |x|}
    \notag \\
  &\leq e^{-m_{x}(y) - \frac{C}{2} \log |x|}, \notag
\end{align}
so again (\ref{E:Auvbound}) is valid.

\emph{Case 6}.  $0, y \in S_{v}$.  Here, from
(\ref{E:BSsep}) and (\ref{E:Svar}),
\begin{align}
  P(A(u,v)) &\leq P(0 \lra u) \notag \\
  &\leq e^{-m(u)} \notag \\
  &\leq e^{-2c_{7}C \log |x|}
    \notag \\
  &\leq e^{-m_{x}(y) - c_{7}C \log |x|}, \notag
\end{align}
so once more (\ref{E:Auvbound}) is valid.

Thus (\ref{E:Auvbound}) is valid for all $u, v \in E$
with $s_{x}(v - u) \geq C \log |x|$.  Summing
over such $u, v$ and combining with (\ref{E:reachE}) 
yields (\ref{E:cost}).
\end{proof}

As discussed in the remarks preceding Lemma \ref{L:uncleancost1},
an unclean open path from 0 to some $y$ 
need not cost more than a clean one, outside of the
circumstances of (\ref{E:cost}).  The next lemma shows that when
\emph{all} paths to $y$, open or not, are unclean, then every open
path to $y$ must be as in (\ref{E:cost}), so an extra cost is always 
paid.  The lemma is valid for more general $G$ than stated, but we only 
need the halfspace case.

\begin{lemma} \label{L:uncleancost2}
Assume (\ref{E:assump}).
There exist $C_{i}$ such that if $G$ is either $\Zd$ or the
intersection of a halfspace with $\Zd$, $C \geq C_{11}, |x| \geq C_{12}$ 
and $y \in  Q_{x}(C)$ is not $(x,C)$-cleanly
reachable from 0 inside $G$, then
\[
  P(0 \lra y \text{ in } \mathcal{B}(G)) \leq e^{-m_{x}(y) - C_{10}C \log |x|}.
\]
\end{lemma}
\begin{proof}
Suppose $y$ is not $(x,C)$-cleanly reachable from 0 but
$0 \lra y$ via some path $\gamma$ of open bonds
in $\mathcal{B}(G)$.  (If there
is more than one such $\gamma$, we can choose one arbitrarily.)
Define $0 = w_{0}, w_{1},.., w_{m} = y$ inductively as 
follows: $w_{i+1}$ is the first site in 
$\gamma$ after $w_{i}$ for which $\gamma[w_{i+1},z]
\subset B(w_{i},C_{15}C \log |x|)^{c}$, where $C_{15}$ 
is a constant to be specified; if there is no such $w_{i+1}$ for 
some value $i = m-1$, then $y \in B(w_{i},
C_{15}C \log |x|)$ and we end the construction.  Let
$B_{i} = B(w_{i},
C_{15}C \log |x|)$.

Since $w_{i+1} \in \partial B_{i}$, it is easy to see that 
there exists a lattice path $\alpha$ from 0 to 
$y$ in $\mathcal{B}(G)$ contained in $\cup_{i=0}^{m} B_{i}$, 
and $\alpha$ can be chosen so that once $\alpha$
leaves any of the balls $B_{i}$, 
it does 
not return to $\cup_{j \leq i} B_{j}$.
(Since $G \cap B_{i}$ is ``connected'' for each $i$, 
one need only ensure that when $\alpha$ exits any ball
$B_{i}$, it exits into the ball $B_{j}$ of maximal index $j$
for which $B_{j} \cap B_{i} \neq \phi$.  Informally speaking,
$\alpha$ is approximately $\gamma$ with doublebacks erased.)
Since $y$ is not $(x,C)$-cleanly reachable
from 0 inside $G$, there must exist sites $u, v$ in $\alpha$ 
with $u$ preceding 
$v$ such that
$s_{x}(v - u) \geq C \log |x|$.  Let $k$ and $l$ be such that
$u \in B_{k}, v \in B_{l}$.
Provided $C_{15}$ is small enough, we have by Lemma
\ref{L:basicprops}(ii):
\[
  |s_{x}(u - w_{k})| \leq \frac{1}{4} C \log |x|, \qquad
  |s_{x}(v - w_{l})| \leq \frac{1}{4} C \log |x|
\]
and therefore 
\begin{equation} \label{E:sxwjwi}
  s_{x}(w_{l} - w_{k}) \geq \frac{1}{2} C \log |x|,
\end{equation}
which by Lemma \ref{L:basicprops}(ii) implies $|w_{l} - w_{k}|
> 2C_{15}C \log |x|$.  Thus 
$B_{k}$ and $B_{l}$ 
are disjoint.  Since
$\alpha$ visits $u$ before $v$ and does not visit $\cup_{j \leq k} 
B_{j}$ after 
leaving $B_{k}$, it follows that $k < l$ and so $w_{k}$ precedes 
$w_{l}$ in $\gamma$.  But now we are in the situation of Lemma
\ref{L:uncleancost1} (with $b = C_{15}$, $a = 2dM_{0}/m_{0}$ obtained from
Lemma \ref{L:basicprops}(i), and $C/2$ in place of 
$C$):  we have an open path $\gamma$ from 0 to $y$
which visits $w_{k}$, then $w_{l}$, after which it does not return to 
$B_{k}$, and (\ref{E:sxwjwi}) holds; provided $C_{15}$ is small
enough, the lemma follows.
\end{proof}

Let $m_{x}^{+}(\cdot) = \max (m_{x}(\cdot),0)$.

\begin{lemma} \label{L:extracost}
  Assume (\ref{E:assump}).
  There exist constants $C_{i}$ such that if $D \subset \Zd,
  C \geq 1, |x| \geq C_{13}(C)$ and $v, w \in \tQxC$, then
  \begin{equation} \label{E:xtracost}
    P(v \in \Delta_{x,C}(0,D) \ \text{and} \ 0 \lra w )
    \leq e^{-m_{x}^{+}(w) - C_{12}C \log |x|}.
  \end{equation}
\end{lemma}

Note that, in contrast to (\ref{E:xtracost}), from (\ref{E:mhcompar}) 
and (\ref{E:mxbound}) one obtains that the probability of the event
$0 \lra w$ alone can be bounded above by $e^{-m(w)}$ or 
$e^{-m_{x}^{+}(w)}$.  The significance of Lemma \ref{L:extracost}
is that because $v \in \Delta_{x,C}(0,D)$, so that $v$ is barely cleanly 
reachable from 0, the additional presence of a path $0 \lra v$ introduces
an extra cost of $C_{12}C \log |x|$, even though the two paths 
need not be disjoint.

\begin{proof}[Proof of Lemma \ref{L:extracost}]
There exist $C_{16}(C)$ and $c_{15}$ such that $|x| \geq C_{16}$ and 
$w \in \tQxC$ imply $|w| \leq c_{15}|x|$.  Hence as in 
(\ref{E:reachE}), we have for some $c_{16}$:
\begin{equation}  \label{E:reachball2}
  P(0 \lra \partial B(0,c_{16}|x|)) \leq 
  e^{-m_{x}(w) - C \log |x|}.
\end{equation}
Let $0 < c_{17} < c_{18}$ be constants to be specified later, let
\[
  E = B(0,c_{16}|x|), \qquad B_{v} = B
  (v,4dc_{18}C \log |x|), \qquad \tilde{B}_{v} = 
  B(v,2dc_{18}C \log |x|),
\]
and let $N$ denote the event that there is a $(c_{17}C \log |x|)$-near
connection from 0 to $w$ in $\mathcal{B}(E
\backslash B_{v})$.
If $c_{17}$ is sufficiently small relative to $c_{18}$, and $|x|$ and $C$ 
are sufficiently large, then by Lemmas \ref{L:rweakmix}, 
\ref{L:reachball} and \ref{L:rnear} we have
{\allowdisplaybreaks
\begin{align} \label{E:ifrnear}
  P\bigl([v &\in \Delta_{x,C}(0,D)] \cap N \bigr) \\
  &\leq P([v \lra \partial \tilde{B}_{v}] \cap N) \notag \\
  &\leq 2P(v \lra \partial \tilde{B}_{v}) P(N) \notag \\
  &\leq 2 e^{-m_{0}c_{18}C \log |x|} e^{-m_{x}(w)+
    C_{5}c_{17}C \log |x|} \notag \\
  &\leq 2 e^{-m_{x}(w) - c_{19}C \log |x|}.\notag
\end{align}
}% end of \allow
Next, we suppose first that $w \notin \overline{B_{v}}$.
Using Lemma \ref{L:rweakmix}, again assuming $|x|$ and $C$ are 
sufficiently large,
{\allowdisplaybreaks
\begin{align} \label{E:nornear}
  P\bigl(&[v \in \Delta_{x,C}(0,D)] \cap [0 \lra w] \cap N^{c} \bigr) \\
  &\leq P(0 \lra \partial E) + \sum_{I,J} P\bigl(\Gamma(0,E \backslash
    B_{v}) = I, \Gamma(w,E \backslash B_{v}) = J \bigr) \notag \\
  &\leq P(0 \lra \partial E) + \sum_{I,J} 2P\bigl(\Gamma(0,E \backslash
    B_{v}) = I \bigr)P\bigl(\Gamma(w,E \backslash B_{v}) = J \bigr) \notag \\
  &\leq P(0 \lra \partial E) + 
    2P\bigl(0 \lra \partial B_{v} \text{ in } \mathcal{B}(\tQxC) \bigr)
    P(\partial B_{v} \lra w ), \notag
\end{align}
}%end of allow
where the sum is over all $I, J$ with $0 \in I, w \in J, 
J \cap \partial B_{v} \neq \phi, d(I,J) > c_{17}C
\log |x|$ and with $I$ containing the sites of a path from 0 to 
$\partial B_{v}$ in $\mathcal{B}(\tQxC)$.
Presuming $c_{18}$ is sufficiently small we have 
$s_{x}(u - z) \leq (C \log |x|)/4$ for all $u, z \in \overline{B_{v}}$.  
Since $v$ is barely $(x,C)$-cleanly reachable from 0 inside $G_{x}$, 
it follows readily that no $y \in \partial B_{v}$ is $(x,C/2)$-cleanly
reachable from 0 inside $G_{x}$.  Hence by Lemma \ref{L:uncleancost2},
for some $c_{20} < 1$,
\begin{equation} \label{E:costofy}
  P\bigl(0\lra y \text{ in } \mathcal{B}(\tQxC) \bigr) \leq 
  e^{-m_{x}(y) - 5c_{20}C \log |x|} \qquad \text{for all}
  \ y \in \partial B_{v}.
\end{equation}
Further, if $|x|$ is large and $c_{18}$ is 
chosen sufficiently small, for $y \in \overline{B_{v}}$
we have from (\ref{E:mxbound})
and (\ref{E:mbounds}):
\begin{equation} \label{E:mxdiff}
  |m_{x}(y) - m_{x}(v)|
  \leq c_{20}C \log |x|.
\end{equation}
Therefore if $|x|$ is sufficiently large, (\ref{E:costofy}) yields
\begin{equation} \label{E:costofy2}
  2P\bigl(0\lra \partial B_{v} \text{ in } \mathcal{B}(\tQxC) \bigr) \leq 
  2|\partial B_{v}| 
  e^{-m_{x}(v) - 4c_{20}C \log |x|} \leq 
  e^{-m_{x}(v) - 3c_{20}C \log |x|}
\end{equation}
and similarly
\begin{equation} \label{E:costofvw}
  P(\partial B_{v} \lra w) \leq e^{-m_{x}(w - v) + 2c_{20}C \log |x|}.
\end{equation}
If $w \in \overline{B_{v}}$, the same argument applies with $P(\partial B_{v}
\lra w)$ replaced by 1 throughout, by (\ref{E:mxdiff}).
Combining (\ref{E:reachball2}), (\ref{E:nornear}), (\ref{E:costofy2})
and  (\ref{E:costofvw}) shows that 
\[
  P\bigl([v \in \Delta_{x,C}(0,D)] \cap [0 \lra w] \cap N^{c} \bigr) 
  \leq 3 e^{-m_{x}(w) - c_{20}C \log |x|}, 
\]
which with (\ref{E:ifrnear}) proves (\ref{E:xtracost}) with $m_{x}$
in place of $m_{x}^{+}$.  But since $v$ is barely $(x,C)$-cleanly
reachable from 0 inside $G_{x}$, for some $c_{21}$ and $c_{22}$ 
we must have $|v| \geq c_{21}C \log |x|$ and hence the left side of 
(\ref{E:xtracost}) is bounded by 
\[
  P(0 \lra v) \leq e^{-c_{22}C \log |x|},
\]
so we can replace $m_{x}$ with $m_{x}^{+}$ in (\ref{E:xtracost}).
\end{proof}

Let $\mathcal{S}(C,r,x)$ denote the set of all gapped 
$(C,r,x)$-skeletons derived from all paths $\gamma$ (starting from
0) in all configurations $\omega$, and
\begin{align}
  \mathcal{S}_{jl}(C,r,x) = \{(u_{i},&v_{i},v_{i}^{\prime})_{i \leq k}
    \in \mathcal{S}(C,r,x):  v_{k} = z, \notag \\
  &|S((u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k})| = j,
    |L((u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k})| = l \}. \notag 
\end{align}

The next result is the analog of Lemma 2.3 of \cite{Al1}.

\begin{lemma} \label{L:atmost3n}
  Assume (\ref{E:assump}).
  There exist constants $C_{i} > 3$ such that if $C \geq C_{13},
  r \geq C_{14}$ and $|x| \geq C_{17}(C)$, then for $n$ sufficiently
  large, there exist a configuration $\omega$ and a path $\gamma$
  from 0 to $nx$ for which the gapped $(C,r,x)$-skeleton consists
  of at most $3n$ tuples.
\end{lemma}
\begin{proof}
Fix $x \in \Zd$ and $C > 1$.  The conclusion will follow if we can show 
that
\begin{align}  \label{E:goal}
  P(0 \lra nx \ &\text{via a path} \ \gamma \ \text{for which the gapped }
    (C,r,x)-\text{skeleton consists of} \\
  &\text{more than} \ 3n \ \text{tuples}) < P(0 \lra nx), \quad \text{for}
    \ n \ \text{large}. \notag
\end{align}
 From the definition of $m(x)$ we have
\begin{equation} \label{E:nxlowerbound}
  P(0 \lra nx) \geq 2^{-n}e^{-nm(x)} \quad \text{for} \ n \ \text{large}.
\end{equation}
Fix $j, l$ and $U = (u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k}
\in \mathcal{S}_{jl}(C,r,x,z).$  Let $\Lambda_{i}, 0 \leq i \leq k$, be
subsets of $\mathcal{B}(\Zd)$ which are possible values of the clusters
$\Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i})$, subject to (\ref{E:prop2})
and satisfying $v_{i}, w_{i} \in \Lambda_{i}$ whenever $\Lambda_{i}
\neq \phi$, where $D_{i} = \Zd \backslash
\cup_{j < i}(\Lambda_{j})^{r \log |x|}$ as in Definition \ref{D:skel}.
That is, we suppose there exists a configuation $\omega$ for which
$v_{i},w_{i} \in \Lambda_{i}$ whenever $\Lambda_{i} \neq \phi$,
and $\Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i},\omega) = \Lambda_{i}$
for all $i \leq k$, and (\ref{E:uwdist}) holds.  (Note that we can have
$\Lambda_{i} = \phi$ only for $i = k$.)  We call such sequences of sets
$\Lambda_{i}, 0 \leq i \leq k$, \emph{allowable}.  Then
\begin{align} \label{E:bound1}
  P&\bigl(0 \lra nx \ \text{via a path} \ \gamma \ \text{with gapped}
    \ (C,r,x)-\text{skeleton} \ U \\
  &\qquad \text{and} \ \Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i}) =
    \Lambda_{i} \ \text{for all } i \leq k \bigr) \notag \\
  &\leq P\bigl( \Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i}) =
    \Lambda_{i} \ \text{and} \ \langle v_{i}v_{i}^{\prime} \rangle
    \ \text{is open for all} \ i \leq k \bigr). \notag
\end{align}
Here, for convenience of notation, we define the event ``$\langle v_{k}
v_{k}^{\prime} \rangle$ is open'' to be the full probability space
$\{0,1\}^{\mathcal{B}(\Zd)}$, since $v_{k} = v_{k}^{\prime} = z$.
If $|x| \geq C_{1}(C)$ then by (\ref{E:tqq}) and Lemma
\ref{L:basicprops}(i),
\begin{equation} \label{E:diambound}
  \diam(\overline{\Lambda}_{i}) \leq
  \diam(Q_{x}(C)) + 2 \leq c_{23}|x|.
\end{equation}
Since the event $[\Gamma(u_{i},(u_{i} + \tQxC) \cap D_{i}) =
\Lambda_{i}$ and $\langle v_{i}v_{i}^{\prime} \rangle$ is open$]
\in \mathcal{F}_{\overline{\Lambda}_{i}}$, if $|x|$ and $r$ are
sufficiently large then from (\ref{E:prop2}), (\ref{E:diambound})
and Lemma \ref{L:rweakmix} it follows that
\begin{align}  \label{E:rweakapp}
  P\bigl(\Gamma &(u_{i},(u_{i} + \tQxC) \cap D_{i}) =
    \Lambda_{i} \ \text{and} \ \langle v_{i}v_{i}^{\prime} \rangle \
    \text{is open for all} \ i \leq k \bigr) \\
  &\leq 2^{k} \prod_{i \leq k} P\bigl(\Gamma (u_{i},(u_{i} + \tQxC)
    \cap D_{i}) =
    \Lambda_{i} \ \text{and} \ \langle v_{i}v_{i}^{\prime} \rangle \
    \text{is open} \bigr). \notag
\end{align}
 From (\ref{E:tqq}) and Lemma \ref{L:basicprops}(i),
for a given value of $u_{i}$ there are at most $c_{24}|x|^{d}$ choices
for each of $v_{i},v_{i}^{\prime}$ and $w_{i}$, for some $c_{24}(C)$.
Therefore for some $c_{25}$, if $|x| \geq C_{1}(C)$ then
\begin{equation} \label{E:Sbound}
  |\mathcal{S}_{jl}(C,r,x,nx)| \leq e^{c_{25}(j+l) \log |x|}.
\end{equation}
If $C$ is large enough and $|x| \geq c_{26}(C)$ then by (\ref{E:tqq}),
(\ref{E:bound1}), (\ref{E:diambound}), Lemma \ref{L:extracost}
and Lemma \ref{L:basicprops}(iii), summing (\ref{E:rweakapp}) over
all allowable sequences $\{\Lambda_{i}, 0 \leq i \leq k\}$ gives
{\allowdisplaybreaks
\begin{align}  \label{E:bound2}
  P\bigl(0 &\lra nx \ \text{via a path} \ \gamma \ \text{with gapped }
    (C,r,x)-\text{skeleton} \ U \bigr) \\
  &\leq 2^{k} \prod_{i \leq k} P\bigl(v_{i} \in \Delta_{x,C}(u_{i},D_{i})
    \cup (u_{i} + \partial_{in}G_{x}); \notag \\*
  &\qquad \qquad \qquad u_{i} \lra v_{i} \ \text{and} \ u_{i} \lra w_{i} \
    \text{both in} \ \mathcal{B}(u_{i} + \tQxC)) \notag \\
  &\leq 2^{k} \exp\left(-\sum_{i \in S(U)} m_{x}^{+}(w_{i} - u_{i})
    - C_{12}C|S(U)|\log |x| \right) \notag \\*
  &\qquad \times \exp\left(-\max\left[\sum_{i \notin S(U)} m_{x}(w_{i}
    - u_{i}),\sum_{i \in L(U)} m_{x}(v_{i} - u_{i}) \right] \right) \notag \\
  &\leq 2^{j+l} \exp\left(-\sum_{i \in S(U)} m_{x}^{+}(w_{i} - u_{i})
    - C_{12}Cj \log |x| \right) \notag \\*
  &\qquad \times \exp\left(-\max\left[\sum_{i \notin S(U)} m_{x}(w_{i}
    - u_{i}),l(m(x) - M_{0}) \right] \right). \notag
\end{align}
}%end of \allow
The remainder of the proof follows that of Lemma 2.3 of \cite{Al1}.
(It should be noted at this point that there is a significant misprint in
that proof, corrected in Remark \ref{R:correc} below.)  Choose $C$ 
such that $C_{12}C\geq 4c_{25} + 6dM_{0}r$. We consider
first $l \geq 3n$.  If $|x|$ is large, then using (\ref{E:bound2}),
(\ref{E:Sbound}) and (\ref{E:nxlowerbound}),
\begin{align} \label{E:3nplus}
  \sum_{l \geq 3n} &\sum_{j \geq 0} P\bigl(0 \lra nx \ \text{via a path}
    \ \gamma \ \text{with gapped} \ (C,r,x)-\text{skeleton in} \
    \mathcal{S}_{jl}(C,r,x,nx) \bigr) \\
  &\leq \sum_{l \geq 3n} \sum_{j \geq 0} 2^{j+l} e^{c_{25}(j+l)
    \log |x|} e^{-C_{12}Cj \log |x|} e^{-l(m(x) - M_{0})}\notag \\
  &\leq 2 e^{-3n[m(x) - M_{0} - c_{25} \log |x| - \log 2]} \notag \\
  &\leq e^{-2nm(x)} \notag \\
  &= o(P(0 \lra nx)) \quad \text{as} \ n \to \infty. \notag
\end{align}
Next we consider $n \leq l < 3n$.  Again from (\ref{E:bound2}),
(\ref{E:Sbound}) and (\ref{E:nxlowerbound}), for $|x|$ large,
\begin{align} \label{E:nto3n}
  \sum_{n \leq l < 3n} &\sum_{j \geq 3n-l} P\bigl(0 \lra nx \ \text{via a path}
    \ \gamma \ \text{with gapped} \\
  &\qquad \qquad \qquad (C,r,x)-\text{skeleton in} \ 
    \mathcal{S}_{jl}(C,r,x,nx) \bigr) \notag \\
  &\leq \sum_{n \leq l < 3n} \sum_{j \geq 3n-l} 2^{j+l} e^{c_{25}(j+l)
    \log |x|} e^{-C_{12}Cj \log |x|} e^{-l(m(x) - M_{0})}\notag \\
  &\leq 2e^{-2C_{12}Cn \log |x|} \sum_{l \geq n} e^{-l(m(x) 
    - M_{0} - 4c_{25} \log |x|)}\notag \\
  &\leq 2e^{-nm(x) - n(C_{12}C \log |x| - M_{0})} \notag \\
  &= o(P(0 \lra nx)) \quad \text{as} \ n \to \infty. \notag
\end{align}
Finally we consider $l < n$.  From (\ref{E:mxbound}), 
(\ref{E:mbounds}) and (\ref{E:uwdist}) we have
\[
  m_{x}(u_{i+1} - w_{i}) \leq 2dM_{0}r \log |x|
\]
so
\[
  \sum_{i \leq k} m_{x}(w_{i} - u_{i}) = m_{x}(nx) - 
  \sum_{i \leq k-1} m_{x}(u_{i+1} - w_{i}) \geq nm(x) 
  - 2kdM_{0}r \log |x|.
\]
With (\ref{E:bound2}),
(\ref{E:Sbound}) and (\ref{E:nxlowerbound}) this shows
{\allowdisplaybreaks
\begin{align} \label{E:upton}
    \sum_{0 \leq l < n} &\sum_{j \geq 3n-l} P\bigl(0 \lra nx \ \text{via a path}
    \ \gamma \ \text{with gapped} \\*
  &\qquad \qquad \qquad (C,r,x)-\text{skeleton in} \ 
    \mathcal{S}_{jl}(C,r,x,nx) \bigr) \notag \\
  &\leq \sum_{0 \leq l < n } \sum_{j \geq 3n-l} 2^{j+l} e^{c_{25}(j+l)
    \log |x|} e^{-\sum_{i \leq k} m_{x}(w_{i} - u_{i}) 
    -C_{12}Cj \log |x|}\notag \\
  &\leq \sum_{0 \leq l < n } \sum_{j \geq 3n-l} 2^{j+l} e^{c_{25}(j+l)
    \log |x|} e^{-nm(x) + 2(j+l)dM_{0}r \log |x| 
    -C_{12}Cj \log |x|}\notag \\
  &\leq 2n\cdot 2^{3n}e^{3nc_{25} \log |x| - nm(x) + 6ndM_{0}r \log |x| 
    - 2C_{12}Cn \log |x|} \notag \\
  &\leq e^{-nm(x) - C_{12}Cn \log |x|} \notag \\
  &= o(P(0 \lra nx)) \quad \text{as} \ n \to \infty. \notag
\end{align}
}% end of \allow
Statement (\ref{E:goal}) now follows from (\ref{E:3nplus}),
(\ref{E:nto3n}) and (\ref{E:upton}).
\end{proof}

Theorem \ref{T:main}(ii) is an immediate consequence of Theorem 1.9 
of \cite{Al3} and the next Proposition, which proves slightly more than 
CHAP for the function $h$.
\begin{proposition} \label{P:conhull}
  Assume (\ref{E:assump}).  There exist $C$ and $M$ such that
  \begin{equation} \label{E:convhull}
    \frac{x}{\alpha} \in \Co(\tQxC) \quad \text{for some} \ \alpha
    \in [2,6], \quad \text{for all} \ x \in \Zd \ \text{with} \ x \geq M.
  \end{equation}
\end{proposition}
\begin{proof}
In the notation of Lemma \ref{L:atmost3n}, let $C, r$ satisfy
$C \geq C_{13}$ and
$C_{14} \leq r \leq c_{27}C$, where $c_{27}$ is a constant to be 
specified later, and suppose $|x| \geq C_{17}(C)$.  By Lemma 
\ref{L:atmost3n} there exist $n$ and a gapped $(C,r,x)$-skeleton
$(u_{i},v_{i},v_{i}^{\prime},w_{i})_{i \leq k}$ corresponding to
some path from 0 to $nx$, with $k < 3n$.  From (\ref{E:prop1}) we
have $w_{i} - u_{i} \in \tQxC$.  For each $i < k$ there is a path
$\varphi_{i}$ from $w_{i}$ to $u_{i+1}$ of length $|u_{i+1} - 
w_{i}|_{1}$.  If $z,y$ are vertices of $\varphi_{i}$ then by Lemma
\ref{L:basicprops}(ii) and (\ref{E:uwdist}), provided $c_{27}$
is chosen small enough,
\[
  s_{x}(z - y) \leq 2|z - y|_{1} \log \frac{1}{p} \leq 4dr(log |x|) \log 
  \frac{1}{p} \leq C \log |x|,
\]
and by (\ref{E:mxbound}) and (\ref{E:mbounds}), if $|x| \geq
c_{28}(C)$,
\[
  m_{x}(z - y) \leq 2M_{0}|u_{i+1} - w_{i}|_{1} \log \frac{1}{p}
  \leq 4drM_{0}(\log |x|) \log \frac{1}{p} \leq m(x).
\]
If follows that $u_{i+1} - w_{i} \in \tQxC$.  Thus we have
\begin{equation} \label{E:nxrep}
  nx = \sum_{i=0}^{k} (w_{i} - u_{i}) + \sum_{i=0}^{k-1}
  (u_{i+1} - w_{i}) = \sum_{y \in \tQxC} n(y)y,
\end{equation}
where $n(y)$ is the number of times $y$ appears in the first two sums 
in (\ref{E:nxrep}).  Since
\[
  \sum_{y \in \tQxC} n(y) = 2k+1 \in [2n,6n],
\]
the conclusion (\ref{E:convhull}) is obtained by dividing 
(\ref{E:nxrep}) by $\sum_{y \in \tQxC} n(y)$.
\end{proof}

\begin{remark} \label{R:correc}
In the proof of (\cite{Al1} Lemma 2.3), the top three lines on page 1554
should read as follows:

First, for $||x|| \geq$ some $c_{10}$, by (2.12) and Lemma 2.2(ii),
\begin{align}
  \sum_{k \geq 3n} \sum_{j \geq 0} \sum_{(v_{i}) \in 
    \Gamma_{jk}^{x}(nx)} &\left( \prod_{i \notin L((v_{i}))}
    e^{-s_{x}(v_{i+1} - v_{i}) - \sigma g_{x}(v_{i+1} -v_{i})} \right)
    \notag \\
  &\times \left( \prod_{i \in L((v_{i}))} e^{-\sigma g_{x}(v_{i+1} 
    - v_{i})} \right).  \notag
\end{align}
\end{remark}

\section{Proof of Theorem 1.1$(i)$.}
Theorem \ref{T:main}(i) is a consequence of Proposition \ref{P:conhull}
above, together with some results from \cite{Al1} (Lemmas 2.6, 2.8, and 
2.9 and Proposition 2.7) modified only slightly.  Therefore we will give 
only a sketch of the proof.

We say that a path $\gamma \ x$-\emph{backtracks by } $t$ if there exist
sites $u, v$ in $\gamma$ with $u$ preceding $v$ but $m_{x}(v - u) \leq 
-t$.  Since $0 \leq h(v-u) = m_{x}(v - u) + s_{x}(v-u)$, this implies
$s_{x}(v - u) \geq t$.  Thus an $(x,C)$-clean path cannot $x$-backtrack 
by more than $C \log x$.  

By Proposition \ref{P:conhull}, there exists $C$ such that for $|x| \geq M$ 
we can express $x$ as
\[
  x = \sum_{i=1}^{d+1} \alpha_{i}y_{i} \quad \text{with} \ \alpha_{i}
  \geq 0, \quad
  2 \leq \sum_{i=1}^{d+1} \alpha_{i} \leq 6, \quad \text{and} \quad 
  y_{i} \in \tQxC.
\]
In fact, since we can have $y_{i} = y_{j}$ for $i \neq j$, we have a similar 
statement with $\alpha_{i} \leq 1$ for all $i$:
\begin{equation} \label{E:convcomb}
  x = \sum_{i=1}^{d+6} \alpha_{i}y_{i} \quad \text{with} \ 0 \leq 
  \alpha_{i} \leq 1 \quad \text{and} \ y_{i} \in \tQxC.
\end{equation}
For each $y_{i}$, there is an $(x,C)$-clean path from 0 to $y_{i}$, and 
consequently $s_{x}(y_{i}) \leq C \log x$.  We would like to find a constant
$b$, depending only on $P$, such that
\begin{equation} \label{E:sxbound}
  s_{x}(\alpha_{i}y_{i}) \leq bC \log |x| \quad \text{for all} \ i.
\end{equation}
Of course $\alpha_{i}y_{i}$ need not be in $\Zd$, in which case $s_{x}
(\alpha_{i}y_{i})$ is not defined, but we can replace $\alpha_{i}y_{i}$ 
with an ``adjacent'' lattice site at the expense of an 
easily manageable error.  If we 
can establish (\ref{E:sxbound}), then (\ref{E:convcomb}) and subadditivity
of $s_{x}$ give
\begin{equation} \label{E:sxxbound}
  s_{x}(x) \leq  \sum_{i=1}^{d+6} \alpha_{i}y_{i} \leq (d+6)bC \log |x|.
\end{equation}
Since $m_{x}(x) = m(x) = m(\theta) |x|$, we have $P(0 \lra x) = 
e^{-m(\theta)|x| - s_{x}(x)}$, so (\ref{E:sxxbound}) yields the desired
conclusion (\ref{E:23bound}).  Thus it is enough to prove the following:
\begin{equation} \label{E:yalphay}
  \text{there exists} \ b \ \text{such that if} \ y \in 
  \tQxC \ \text{and} \ 0 \leq
  \alpha \leq 1, \ \text{then} \ s_{x}(\alpha y) \leq bC \log |x|.
\end{equation}
(Again, $\alpha y$ should be interpreted as an adjacent lattice site if 
$\alpha y$ is not itself a lattice site.)  To prove (\ref{E:yalphay}), let 
$y \in \tQxC$ and $0 \leq \alpha \leq 1$, and let $\gamma: [0,1] \to 
\mathbb{R}^{d}$ be an $(x,C)$-clean lattice path from 0 to $y$.
Since $\gamma$ does not $x$-backtrack by $C \log |x|$ or more, 
we can approximate $\gamma$ to within $C \log |x|$ (measured in 
the norm $m(\cdot)$) by a curve $\tilde{\gamma}$ (not necessarily a
lattice path) from 0 to $y$ which does not $x$-backtrack at all; in
fact we can have $m_{x}(\tilde{\gamma}(t))$ strictly increasing.
Proposition 2.7 of \cite{Al1} then states that there exist $k_{d}$, 
depending only on the dimension $d = 2$ or 3, and a collection of 
$k_{d}$ subintervals $[s_{j},t_{j}], j = 1,..,k_{d}$, of [0,1], such 
that 
\begin{align} \label{E:alphayrep}
  \alpha y &=  \sum_{i=1}^{k_{d}} (\tilde{\gamma}(t_{j}) - 
    \tilde{\gamma}(s_{j})) \\
  &= \sum_{i=1}^{k_{d}} [(\tilde{\gamma}(t_{j}) - 
    \gamma(t_{j})) + (\gamma(t_{j}) - \gamma(s_{j}))
    + (\gamma(s_{j}) - \tilde{\gamma}(s_{j}))]. \notag 
\end{align}
(The intervals $[s_{j},t_{j}]$ may depend on $\gamma$ here, and may
overlap.)
Let us assume all the points appearing in (\ref{E:alphayrep}) are lattice
sites; otherwise we again approximate them by adjacent lattice sites.
Since $\gamma$ is $(x,C)$-clean, we have
\[
  s_{x}(\gamma(t_{j}) - \gamma(s_{j})) \leq C \log |x|.
\]
 From (\ref{E:mbounds}) and Lemma \ref{L:basicprops}(ii), for some 
$c_{29}$ depending only on $P$,
\[
  s_{x}(\tilde{\gamma}(t_{j}) - \gamma(t_{j})) \leq c_{29}
  m(\tilde{\gamma}(t_{j}) - \gamma(t_{j})) \leq c_{29}C \log |x|
\]
and similarly for $s_{x}(\tilde{\gamma}(s_{j}) - \gamma(s_{j}))$.
Together with (\ref{E:alphayrep}), these bounds yield
\[
  s_{x}(\alpha y) \leq k_{d}(4c_{29} + 1)C \log |x|,
\]
so (\ref{E:yalphay}) is proved.

\section{Proof of Theorem 1.4}
We will adapt the techniques of (\cite{Al1}, Lemma 4.3) to our 
present context.

We may assume that $0 \in \partial H$.
Let $n$ denote the outward unit normal to $H$.  Given 
$\omega$ in which $0 \leftrightarrow x$, let $\gamma_{\omega}$ be 
an open path in $\omega$ from $0$ to $x$ (chosen arbitrarily if there
is more than one such path) and let $X(\omega)$ be a site in 
$\gamma_{\omega}$ 
which maximizes
$n \cdot y$ over $y \in \gamma_{\omega}$ (the first
such site, say, 
if there is more than one.) Note that if the segment 
of $\gamma_{\omega}$ from $0$ to $X(\omega)$ and the segment
from $X(\omega)$ to $x$ are interchanged, the result is a path from
$0$ to $x$ in $H$.  To make such an interchange
possible, in a sense, we must show that the two segments are 
nearly independent.

By Lemma 2.7, provided $|x|$ is large and $c_{30} \geq C_{4}$, the event
\[
  N = \{\omega: \ \text{there is a} \ (c_{30} \log |x|) \text{-near 
  connection from} \ 0 \ \text{to} \ x \ \text{in} \ \omega\}
\]
satisfies 
\begin{equation} \label{E:Nbound}
  P(N) \leq e^{-m(x) + c_{30}C_{5}\log |x|}.
\end{equation}
By Lemma \ref{L:reachball} and Theorem \ref{T:main},
there exists $c_{31}$ 
such that for $B_{0} = B(0,c_{31}|x|)$ the event
\[
  U = \{\omega: 0 \leftrightarrow \partial B_{0}\}
\]
satisfies
\begin{equation} \label{E:Ubound}
  P(U) \leq \frac{1}{2}P(0 \leftrightarrow x).
\end{equation}
Therefore there exist $z \in B_{0}$
and $c_{32}$ such that, letting $H_{z}$ denote the
translate of $H$ with $z \in \partial H_{z}$,
\begin{equation} \label{E:zprop}
  P(0 \leftrightarrow z \leftrightarrow x
  \ \text{in} \ \mathcal{B}(H_{z} \cap B_{0})) \geq
  P(0 \leftrightarrow x, X = z, U^{c})
  \geq \frac{c_{32}}{|x|^{d}} P(0 \leftrightarrow x).
\end{equation}
Note that $z$ is not in the interior of $H$, so $x-z \in H$.
Using positive connection correlations,
\begin{align}
  P(0 \leftrightarrow x \ \text{in} \ \mathcal{B}(H))
    &\geq P(0 \leftrightarrow x - z \leftrightarrow x
    \ \text{in} \ H) \notag \\
  &\geq P(0 \leftrightarrow x-z \ \text{in} \  \mathcal{B}(H))
    P(x-z \leftrightarrow x \ \text{in} \  \mathcal{B}(H)) \label{E:split} \\
  &= P(0 \leftrightarrow z \ \text{in} \  \mathcal{B}(H_{z}))
    P(z \leftrightarrow x \ \text{in} \  \mathcal{B}(H_{z})). \notag
\end{align}

We need to compare the left side of (\ref{E:zprop}) to the right
side of (\ref{E:split}).  For $x \in \Zd$ let $f_{d}(x)$ be
$\log |x|$ for $d = 2,3$
and $(\log |x|)^{2}$ for $d \geq 4$.
By Lemma \ref{L:reachball}, there 
exists $c_{33}$ such that, defining
\[
  B_{z} = B(z,2 + c_{33} f_{d}(x)), \qquad 
  \tilde{B}_{z} = B(z,c_{33} f_{d}(x)),
\]
we have
\begin{equation} \label{E:reach}
  P(z \leftrightarrow \partial \tilde{B}_{z})
  \leq \frac{c_{32}A}{4}
  e^{-(d+c_{30}C_{5}+C) f_{d}(x)},
\end{equation}
where $A, C$ are as in Theorem \ref{T:main}.
($A = 1$ if $d \geq 4$.)  Assume first that
$0, x \notin B_{z}$.
Define
\[
  \tilde{N} = \{\omega: \ \text{there is a} \ (c_{30} \log |x|) 
  \text{-near 
  connection from} \ 0 \ \text{to} \ x \ \text{outside} \ B_{z}
  \ \text{in} \ \omega\}.
\]
For $|x|$ large, Lemma \ref{L:rweakmix}, (\ref{E:reach}),
(\ref{E:Nbound}), Theorem \ref{T:main} and (\ref{E:zprop}) give
\begin{align} \label{E:Qsmall}
  P([0 &\leftrightarrow z \leftrightarrow x
    \ \text{in } H_{z}] \cap \tilde{N}) \\
  &\leq P([z \leftrightarrow \partial \tilde{B}_{z}] \cap
    \tilde{N}) \notag \\
  &\leq 2P(z \leftrightarrow \partial \tilde{B}_{z})
    P(\tilde{N}) \notag \\
  &\leq \frac{c_{32}A}{2}
    e^{-m(x) - (d+C) f_{d}(x)} \notag \\
  &\leq \frac{1}{2} P(0 \leftrightarrow z 
    \leftrightarrow x
    \ \text{in } H_{z} \cap B_{0}). \notag 
\end{align}
Therefore by Lemma \ref{L:rweakmix}, positive connection
correlations and (\ref{E:crudelower}),
for some $c_{34}$, provided $c_{30}$ and $|x|$ are large,
\begin{align} \label{E:addinN}
  P\bigl(0 &\leftrightarrow z 
    \leftrightarrow x
    \ \text{in} \ \mathcal{B}(H_{z} \cap B_{0}) \bigr) \\
  &\leq 2P\bigl([0 \leftrightarrow z 
    \leftrightarrow x
    \ \text{in} \ \mathcal{B}(H_{z} \cap B_{0})] \cap 
    \tilde{N}^{c} \bigr) \notag \\
  &\leq 2P\bigl([0 \leftrightarrow \partial B_{z}
    \ \text{in} \ H_{z} \cap B_{0}]
    \cap [x \leftrightarrow \partial B_{z}
    \ \text{in} \ H_{z} \cap B_{0}]
    \cap \tilde{N}^{c} \bigr) \notag  \\
  &\leq \sum_{I,J} P\bigl(\Gamma(0,(H_{z} \cap B_{0})
    \backslash B_{z}) = I, \Gamma(x,(H_{z} \cap B_{0})
    \backslash B_{z}) = J \bigr) \notag \\
  &\leq \sum_{I,J} 2 P\bigl(\Gamma(0,(H_{z} \cap B_{0})
    \backslash B_{z}) = I \bigr) P\bigl(\Gamma(x,(H_{z} \cap B_{0})
    \backslash B_{z}) = J \bigr) \notag \\
  &\leq 2P\bigl(0 \lra \partial B_{z} \text{ in } \mathcal{B}(H_{z}) \bigr) 
    P\bigl(x \lra \partial B_{z} \text{ in } \mathcal{B}(H_{z}) \bigr) \notag \\
  &\leq \sum_{q,r \in H_{z} \cap \partial B_{z}} 2P(0 \lra q \text{ in } 
    \mathcal{B}(H_{z}))
    P(x \lra r \text{ in } \mathcal{B}(H_{z})) \notag \\
  &\leq e^{c_{34} f_{d}(x)} P(0 \lra z \text{ in } \mathcal{B}(H_{z}))
    P(z \lra x \text{ in } \mathcal{B}(H_{z})) \notag
\end{align}
where the sum is over $I, J \subset (B_{0} \cap H_{z})
\backslash B_{z}$ with $0 \in I, 
I \cap \partial B_{z} \neq \phi, x \in J, J \cap \partial B_{z} \neq
\phi$, and $d(I,J) \geq c_{30} \log |x|$.
Combining (\ref{E:zprop}),
(\ref{E:split}) and (\ref{E:addinN}) yields
\begin{equation} \label{E:halffull}
  P(0 \leftrightarrow x \ \text{in} \  \mathcal{B}(H)) \geq
  e^{-c_{34}f_{d}(x)} \frac{c_{32}}{|x|^{d}} P(0 \leftrightarrow x).
\end{equation}
With Theorem \ref{T:main} this completes the proof, when
$0 \notin B_{z}$ and $x \notin B_{z}$.

When $0 \in B_{z}$ the proof is simpler.  We have
\begin{equation} \label{E:centercon2}
  P(0 \leftrightarrow z \ \text{in} \  \mathcal{B}(H_{z}))
  \geq e^{-c_{35} f_{d}(x)}
\end{equation}
and in place of (\ref{E:addinN}),
\begin{align} \label{E:nodecomp}
  e^{-c_{36} f_{d}(x)}&P(0 \leftrightarrow z 
    \leftrightarrow x
    \ \text{in} \ \mathcal{B}(H_{z} \cap B_{0})) \\
  &\leq P(0 \leftrightarrow 
    z \ \text{in} \ \mathcal{B}(H_{z})) P(z \leftrightarrow 
    x \ \text{in} \ \mathcal{B}(H_{z})). \notag 
\end{align}
Combining (\ref{E:zprop}), (\ref{E:split}), (\ref{E:centercon2})
and (\ref{E:nodecomp}) again yields (\ref{E:halffull}).  The proof
when $x \in B_{z}$ is similar.

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\end{document}