0 \qquad
\forall\,1\leq i \leq p. \ee Indeed, by (\ref{prpol}) we have
\be{HLiineq2} H_{\L_i}(\h) \geq \min_{\xi\in\cV_j} H_{\L_i}(\xi) =
H(\o_j^G) \qquad \mbox{for } j = j(i) =|\L_i\cap\h|. \ee But
$0\leq j0$. Thus, via
(\ref{Hineq}) and (\ref{HLiineq2}), \be{HLiineq3} H(\o_i) - H(\h)
\leq H(\o_i^G) - H(\o_j^G) < H(\o_i^G) \qquad \forall\, 1\leq
i\leq p. \ee Denoting by $\Phi^G(\h,\h')$ the communication height
between $\h$ and $\h'$ for Glauber dynamics, we see that
(\ref{HLiineq3}) in turn yields \be{insys} \Phi^G(\h,\o_p)-H(\eta)
\leq \max\limits_{1\leq i\leq p} H(\o_i) - H(\h) <
\max\limits_{1\leq i\leq p} H(\o_i^G) = \G^G, \ee where $\G^G$ is
the analogue of $\G$ for Glauber. Since $\o_0=\h$ and
$H(\o_p^G)\leq 0$, (\ref{HLiineq3}) gives \be{insys*}
H(\o_p)V} and \ref{pG0} that
\be{narrow}
\cX_{\G_0}=\{\square,\cF(\cX)\},
\ee
because any configuration in $\blacksquare\setminus\cF(\cX)$ is 0-reducible
and therefore cannot be in $\cX_{\G_0}$.
\subsection{Geometric description of minimal saddles and gates}
\label{S2.3}
In this section we develop the focalization property that was
announced in the introduction and we identify a gate for the
nucleation, i.e., a subset of those minimal saddle configurations
the Kawasaki dynamics has to cross with high probability in its
transition from the metastable state $\square$ to the stable state
$\blacksquare$. Our main result is Proposition \ref{C*gate}. In
Section \ref{S2.3.1} we define magic numbers, in Section
\ref{S2.3.2} we study focalization and gates, while in Section
\ref{S2.3.4} we expand the argument slightly further.
\subsubsection{Magic numbers}
\label{S2.3.1}
Let $d(n)\colon\N\to\N_0^3\times \{0,1\}^3$ be the function that associates
with $n$ the unique 6-tuple $(m,l,k,\d,$ $\th,\a)$ appearing in (\ref{ndecomposition2}).
Note that $d(n)$ is a bijection by Proposition \ref{ndecomposition1}.
A special role is played by the set $\bar\N$ of integers $n$ such
that $d(n)=(m,l,0,\d,\th,\a)$. Borrowing terminology from nuclear
physics \cite{P}, we call $\bar\N$ the set of {\it magic numbers}.
For these numbers the principal polyominoes have the form
``quasi-cube + quasi-square''.
If $\bar n\in\bar\N$, then by Theorem \ref{tAC}(b) the associated
(equivalence class of) principal polyominoes $\cR_{\bar l,\bar
l+\bar\a}(\bar m,\bar m+\bar\d, \bar m+\bar\th)$ (defined in
Section \ref{S1.5} item 3) satisfies \be{etahat} \cR_{\bar l,\bar
l+\bar\a}(\bar m,\bar m+\bar\d,\bar m+\bar\th) =
\Big\{\h\in\cV_{\bar n}\colon\; H(\h) =\min_{\h'\in \cV_{\bar n}}
H(\h')\Big\}. \ee
We order $\bar\N$: $\bar\N=\{\bar n_1,\bar n_2,\dots\}$ with $\bar
n_1<\bar n_2<\dots$ Given $\bar n_i\in\bar \N$ with $d(\bar n_i)
=(\bar m_i,\bar l_i,0,\bar\d_i,$ $\bar\th_i, \bar\a_i)$, we have
\be{magsucc} \bar n_{i+1}=\bar n_{i}+1\;\hbox{ if } \bar l_i=0,
\quad\quad \bar n_{i+1}=\bar n_{i}+\bar l_i+\bar\a_i\; \hbox{ if }
\bar l_i\ge 1. \ee
\subsubsection{Focalization and gates}
\label{S2.3.2}
The main result in this section is Proposition \ref{C*gate}. Its
proof relies on the following (recall Section \ref{S1.5} items 3
and 8):
\bp{sellelocali}
Fix $i$ and let $d(\bar n_i)=(\bar m_i,\bar l_i,0,\bar\d_i,\bar\th_i,\bar\a_i)$.\\
(a) Let $\bar n_i$ be such that $\bar l_i+\bar\a_i>1$. Then
$\cD_{\bar l_i,\bar l_i+\bar\a_i}^{2pr,fp}(\bar m_i,\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)
\subseteq\cV_{\bar n_i+2}$ is a gate for $\cV_{\bar n_{i}}\to \cV_{\bar n_{i+1}}$.\\
(b) Let $\bar n_i$ be such that $\bar l_i+\bar \a_i\leq 1$. Then
$\cR^{fp}_{\bar l_i,\bar l_i+\bar\a_i}(\bar m_i,\bar m_i+\bar\d_i,
\bar m_i+\bar\th_i)\subseteq\cV_{\bar n_i+1}$ is a gate for
$\cV_{\bar n_i}\to\cV_{\bar n_{i+1}}$.
\ep
\bpr The proof comes in steps.
\medskip\noindent
{\bf 1.} We begin with the elementary observation that, for all
$n\in\N$ and $\h,\h'\in\cV_n$, \be{gap} H(\h)-H(\h')=kU \mbox{ for
some } k\in\Z, \ee which is immediate from (\ref{Hdef}) and
(\ref{Vn}) since $N_{\L}(\h)=N_{\L}(\h')$.
\medskip\noindent
{\bf 2.} Next, we consider two consecutive magic manifolds.
\bl{l2}
Fix $i$ and let $d(\bar n_i)=(\bar m_i,\bar l_i,0,\bar\d_i,\bar\th_i,\bar\a_i)$.
Then all paths in $(\cV_{\bar n_i}\to\cV_{\bar n_{i+1}})_{opt}$ pass through
$\cR_{\bar l_i,\bar l_i+\bar\a_i}(\bar m_i,\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)
\subseteq\cV_{\bar n_i}$ during the transition from $\cV_{\bar n_i}$
to $\cV_{\bar n_{i}+1}$.
\el
\bpr
Abbreviate $\cR_i=\cR_{\bar l_i,\bar l_i+\bar\a_i}(\bar m_i,\bar m_i+\bar\d_i,
\bar m_i+\bar\th_i)$. Let $\o^K(\bar n_i,\bar n_{i+1})$ be the part of the
reference path $\omega^K$ between the standard polyomino of volume $\bar n_i$
and that of volume $\bar n_{i+1}$.
If $\bar n_i$ is such that $\bar l_i+\bar \a_i>1$, which means that the
corresponding polyomino is neither a quasi-cube nor a quasi-cube plus a
1-protuberance, then by (\ref{magsucc}) we have $\bar n_{i+1}\ge
\bar n_i+2$, and so
\be{camomstar}
\Phi(\cV_{\bar n_{i}},\cV_{\bar n_{i+1}})
\leq \max_{\h\in\o^K(\bar n_i,\bar n_{i+1})} H(\h)
\leq H(\cR_i)+2\D-2U,
\ee
because $2\D$ is the cost to create the $(\bar n_i+1)$-st and the
$(\bar n_i+2)$-nd particle, while $-2U$ is the binding energy when
the $(\bar n_i+1)$-st particle is attached to the droplet.
Suppose that there exists $\o\in(\cV_{\bar n_{i}}\to
\cV_{\bar n_{i+1}})_{opt}$ not passing through $\cR_i$. We will show
that, for any such $\o$,
\be{assurdo}
\max_{\h\in \o } H(\h) \geq H(\cR_i)+\D+U.
\ee
Equations (\ref{camomstar}--\ref{assurdo}) give a contradiction because
$\D+U>2\D-2U$. To prove (\ref{assurdo}), use that $\cR_i=\cF(\cV_{\bar n_i})$
by Theorem \ref{tAC}(b). By (\ref{gap}), if $\o$ does not pass through
$\cR_i$, then the configurations $\h\in\o\cap\cV_{\bar n_{i}}$ have
energy $H(\h)\geq H(\cR_i)+U$. Since the transition from $\cV_{\bar n_{i}}$
to $\cV_{\bar n_{i}+1}$ comes with a further increase in energy by $\D$ due
to the next incoming free particle, it is clear that $\o$ satisfies (\ref{assurdo}).
If $\bar n_i$ is such that $\bar l_i+\bar \a_i\le 1$, then by (\ref{magsucc}) we
have $\bar n_{i+1}=\bar n_i+1$, and so
\be{camomstar1}
\Phi(\cV_{\bar n_{i}},\cV_{\bar n_{i+1}})
\leq \max_{\h\in\o^K(\bar n_i,\bar n_{i+1})} H(\h)
= H(\cR_i)+\D.
\ee
We can now repeat the previous argument via contradiction, because $\D+U>\D$.
\epr
\medskip\noindent
{\bf 3.} We return to the proof of Proposition
\ref{sellelocali}(a,b):
\medskip\noindent
(a) We know from Lemma \ref{l2} that any $\o \in (\cV_{\bar n_i}
\to\cV_{\bar n_{i+1}})_{opt}$ passes through
$\cR_i\subseteq\cV_{\bar n_i}$ and hence crosses $\cV_{\bar
n_{i}+1}$ in $\cR_i^{fp}$. The first non-trivial subsequent move
can only consist in attaching the free particle in $\cR_i^{fp}$ to
the droplet in $\cR_i^{fp}$. Indeed, annihilation of the free
particle would mean to return to $\cV_{\bar n_{i}}$, while without
annihilation the following restrictions are in force:
\begin{itemize}
\item[(i)]
No other free particle can arrive before the free particle is attached,
because otherwise we would have
\be{assurdo3}
\max_{\h\in\o} H(\h) = H(\cR_i) + 2\D
> H(\cR_i) + 2\D -2U\ge \Phi(\cV_{\bar n_{i}},\cV_{\bar n_{i+1}}),
\ee
where the last inequality uses (\ref{camomstar}).
\item[(ii)]
It is not possible to separate a particle from one of the corners of
the droplet before attaching the free particle, since this would increase
the energy by at least $U$ (namely, when the free particle is next to the
site the particle from the corner moves to, the energy increases
by $2U-U=U$), and so we would have
\be{assurdo4}
\max_{\h\in\o} H(\h) \geq H(\cR_i) + \D + U
> H(\cR_i) + 2\D - 2U
\geq \Phi(\cV_{\bar n_{i}},\cV_{\bar n_{i+1}}).
\ee
\end{itemize}
After attaching the free particle we enter the set $\cR_i^{2pr}$, and so
we are in the set
\be{c2u}
\cC_{\h}^{2U}(\cR_i^{2pr})= \Big\{\h'\in\cV_{\bar n_i+1}\colon~
\Phi_{\cV_{\bar n_i+1}}(\h,\h')\leq H(\h)+2U\Big\}
\ee
for some $\h\in\cR^{2pr}_i$.
Now, $\o^K$ visits $\cV_{\bar n_i+1}$ for the last time in $\cF(\cV_{\bar n_i+1})$.
Therefore, by comparison with $\o^K$ and using (\ref{camomstar}), we deduce that
any path in $(\cV_{\bar n_{i} }\to \cV_{\bar n_{i+1} })_{opt}$ has to perform
the passage from $\cV_{\bar n_{i}+1}$ to $\cV_{\bar n_{i} + 2}$ as a single
transition $\h \to \s$ (with $\h\in\cV_{\bar n_i+1}$, $\s\in\cV_{\bar n_i+2}$),
where $\h\in\cF(\cV_{\bar n_{i} +1})$ and $\s$ is obtained from $\h$ by
creating a particle in $\partial^-\L$, so that $H(\s)=H(\h)+\D$.
Moreover, it is clear that any path in $(\cV_{\bar n_{i} }\to\cV_{\bar n_{i+1}})_{opt}$
cannot visit any configuration in $\cV_{\bar n_i+1}$ with energy strictly larger
than $H(\cF(\cV_{\bar n_i+1}))+2U$ since, by (\ref{gap}), this would imply
a value of the energy larger than or equal to $H(\cF(\cV_{\bar n_i+1}))+3U$, which
is strictly larger than the maximal energy in $\o^K$. So, we cannot leave
$\cV_{\bar n_i+1}$ unless we return to $\cV_{\bar n_i}$. This implies that
the transition to $\cV_{\bar n_{i}+2}$ has to be performed through the set
(recall (\ref{Ddef}))
\be{Dunione}
\left(\bigcup_{\h\in\cR_i^{2pr}}
\cF(\cC_{\h}^{2U}(\cR_i^{2pr}))\right)^{fr}
=\cD_{\bar l_i,\bar l_i+\bar\a_i}^{2pr,fp}
(\bar m_i,\bar m_i+\bar\d_i,\bar m_i+\bar\th_i).
\ee
To complete the proof, we note that $\cD_{\bar l_i,\bar l_i+\bar \a_i}^{2pr,fp}
(\bar m_i,\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)\subseteq
\cS(\cV_{\bar n_i},\cV_{\bar n_{i+1}})$ since, by (\ref{ovar}),
\be{sld}
\Phi(\cV_{\bar n_i},\cV_{\bar n_{i+1}})
=\max_s H(\o^K(\bar n_i,\bar n_{i+1}))
= H\left(\cD_{\bar l_i,\bar l_i+\bar\a_i}^{2pr, fp}
(\bar m_i,\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)\right).
\ee
\medskip\noindent
(b) The proof is immediate via Lemma \ref{l2} and the same
reasoning as prior to (\ref{c2u}). \epr
We may now conclude with the main result of this section:
\bp{C*gate}
$\cC^*$ is a gate for $\square\to \blacksquare$.
\ep
\bpr This is immediate from Proposition \ref{V-->V} and
Proposition \ref{sellelocali}(a). \epr
\subsubsection{Further properties of optimal paths between successive
magic manifolds} \label{S2.3.4}
In this section we prove the following extension of Lemma
\ref{l2}.
\bl{l3}
Fix $i$ and let $d(\bar n_i)= (\bar m_i,\bar l_i,0,\bar\d_i,\bar\th_i,\bar\a_i)$
with $\bar l_i\not= 0$. Then all paths in $(\cV_{\bar n_i}\to\cV_{\bar n_{i+1}})_{opt}$
pass through the set obtained from $\cR_{\bar l_i,\bar l_i+\bar \a_i}^{fp}
(\bar m_i,$ $\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)$ by attaching the free
particle to the face of the quasi-cube containing the quasi-square.
\el
\bpr
Again we abbreviate $\cR_i=\cR_{\bar l_i,\bar l_i+\bar\a_i}(\bar m_i,
\bar m_i+\bar\d_i,\bar m_i+\bar\th_i)$.
If $\o\in(\cV_{\bar n_i}\to \cV_{\bar n_{i+1}})_{opt}$, then,
by Lemma \ref{l2}, $\o$ passes through $\cR_i^{fp}\subseteq\cV_{\bar n_i+1}$.
We know from (a) and (b) in the proof of Proposition \ref{sellelocali} that the
creation of another free particle or the separation of a particle from the
droplet are not possible in $\o$ before the free particle is attached. We
want now to prove that if $\o$ passes from $\cR_i^{fp}$ to a configuration
in which the free particle in $\cR_i^{fp}$ is attached to a wrong face of
the droplet (i.e., a face that does not contain the quasi-square) or to the
top of the quasi-square, then $\o$ returns to $\cR_i^{fp}$ before reaching
$\cV_{\bar n_{i+1}}$. This goes as follows.
Let $\cR^{wpr}_i$ be the set of configurations that are obtained
from a configuration in $\cR_i^{fp}$ by attaching the free
particle to a wrong face or to the top of the quasi-square
(leading to a ``wrong protuberance''). We claim that any
configuration $\h'\not\in\cR^{wpr}_i$ that can be obtained from
$\h\in\cR^{wpr}_i$ without again separating the attached particle
from the droplet has an energy $H(\h')\geq H(\h)+2U$, which is
strictly larger than $\Phi(\cV_{\bar n_i},\cV_{\bar n_{i+1}})$. To
see why this claim is true, note that in $\h$ all the particles,
with the exception of the attached one, have at least 3
nearest-neighbors in the cluster. To move a particle of the
quasi-square while keeping it on the same face has energy cost
$\geq 2U$, because the attached particle cannot be a
nearest-neighbor of the site this particle moves to. To move a
particle of the quasi-square away from the face also has energy
cost $\geq 2U$ (possibly with the help of the attached particle
when it is on top of the quasi-square). To move a particle of
another face again has energy cost $\geq 2U$ (possibly with the
help of the attached particle when it is on that same face). Thus,
we would have
\be{assurdo4*}
\max_{\h\in\o} H(\h) = H(\cR_i)+\D-U+2U
> \Phi(\cV_{\bar n_{i}},\cV_{\bar n_{i+1}}),
\ee
where the inequality uses (\ref{camomstar}) and (\ref{camomstar1}). This
contradicts $\o\in(\cV_{\bar n_{i}}\to \cV_{\bar n_{i+1}})_{opt}$.
\epr
Lemma \ref{l3} is only a first step towards describing in more
detail what the optimal paths look like. The problem is to extend
the focalization to manifolds that are not magic, which is
hampered by the degeneracy found in Alonso and Cerf \cite{AC96}
for the isoperimetric inequalities when the number of particles is
not a magic number. At present it seems too difficult to handle
this problem. Some further discussion is provided in Section
\ref{S4.2}.
%%%%%%%%%%% SECTION 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of main theorems}
\label{S3}
\subsection{Proof of Theorem \ref{t1}(a)}
\label{S3.1}
\underline{Upper bound}:
\medskip\noindent
The upper bound on the nucleation time in fact holds uniformly in the
starting point.
\bp{1}
\be{ub}
\lim_{\b\to\infty} \min_{\h\in\cX}
P_\h\Big(\t_{\blacksquare} < e^{(\G+\d)\b}\Big) = 1 \qquad \forall\d>0.
\ee
\ep
\bpr
The proof is achieved by exhibiting a sufficiently probable nucleation event.
\medskip\noindent
{\bf 1.}
Fix $\h\in\cX$ and $\d>0$, put $T=e^{(\G+\frac{\d}{2})\b}$, and define the event
\be{evhit}
\{\t_{\blacksquare}\b_0(\d).
\ee
Putting $T_+=e^{(\G+\d)\b}$ and using (\ref{evlb}) in combination with the
Markov property, we get
\be{evlbit}
\max_{\h\in\cX} P_\h\Big(\t_{\blacksquare} \ge T_+\Big)
\leq \Big(1-e^{-{\d \over 4}\b}\Big)^{T_+/T}
= \Big(1-e^{-{\d \over 4}\b}\Big)^{e^{{\d\over 2}\b}} = \SES.
\ee
\medskip\noindent
{\bf 2.}
To prove (\ref{evlb}), we introduce an event
\be{evdef}
\gE^T_\h\subseteq\{\t_{\blacksquare}0.
\ee
Put $\g=(\G-\G_0)/2$. Then $e^{(\G_0+\g)\b}<{T\over 4}$ for $\b$
sufficiently large and so it follows from (\ref{ubG}) that
\be{evest1}
\min_{\h\in\cX} P_{\h}\left(\t_{\{\square,\blacksquare\}}\leq {T\over 4}\right)
\geq {1 \over 2} \qquad \forall\b>\b_0(\d).
\ee
Thus, it remains to get a lower bound on $r_T$.
\medskip\noindent
{\bf 5.} Let
\be{Asquaredef}
\cA_{\square} = \left\{\h\in\cX\colon\,\exists\,\o\colon\,\h\to\square\colon\,
\max_i H(\o_i)<\G\right\}.
\ee
This set is a cycle with $\cF(\partial\cA_{\square})\supseteq R^{2pr,fp-}$,
the subset of $R^{2pr,fp}$ where the free particle is in the boundary $\partial^-\L$
(because $H(R^{2pr,fp-})=\G$ and removal of the free particle in $R^{2pr,fp-}$
leads to a configuration in $\cA_{\square}$). Hence Proposition \ref{p2.3.2}(a,c) and
(\ref{ubG}) yield
\be{evest2}
P\left(\gE_{ \square}^{R^{2pr,fp},\leq {T\over 4}}\right)
\geq e^{-\d'\b} \qquad \forall\d'>0 \quad \forall \b>\b_0(\d').
\ee
Next, since there is a finite downhill path from $R^{2pr,fp}$ to $R^{spr}$,
there exists a constant $a>0$, independent of $\b$, such that
\be{evest3}
P\left(\gE_{R^{2pr,fp}}^{R^{spr},\leq {T\over 4}}\right) \geq a.
\ee
Next, let
\be{Ablacksquaredef} \cA_{\blacksquare}
= \left\{\h\in\cX\colon\,\exists\,\o\colon\,\h\to\blacksquare\colon\,
\max_i H(\o_i)<\G\right\}.
\ee
This set is a cycle, and $R^{spr}\subseteq\cA_{\blacksquare}$ (as can be
deduced by looking at the reference path $\o^K$ defined in Section \ref{S2.2}).
Since $\square\notin\cA_{\blacksquare}$, Proposition \ref{p2.3.2}(b) and
(\ref{evest1}) yield
\be{evest4}
P\left(\gE_{R^{spr}}^{\blacksquare,\leq {T\over 4}}\right)
\geq {1 \over 2} \qquad \forall \b>\b_0(\d).
\ee
Combining (\ref{stima2}), (\ref{evest2}--\ref{evest3}) and (\ref{evest4})
we find that
\be{stimarT}
r_T \geq {1\over 2} a e^{-\d'\b} \qquad \forall\d'>0 \quad \forall\b>\b_0(\d,\d').
\ee
From (\ref{evdef}), (\ref{stima3}), (\ref{evest1}) and (\ref{stimarT}) we conclude
that,
\be{stima4}
\min_{\h\in\cX} P(\gE^T_\h) \geq e^{-\d''\b} \qquad \forall\d''>0 \quad \forall
\b>\b_0(\d,\d''),
\ee
which proves (\ref{evlb}).
\epr
\medskip\noindent
\underline{Lower bound}:
\bp{lbnuc} $\lim_{\b\to\infty}
P_\square\Big(\t_{\blacksquare}>e^{(\G-\d)\b}\Big) = 1$ for all
$\d>0$. \ep
\bpr We know from Proposition \ref{V-->V} that \be{FVn+2}
\Phi(\square,\blacksquare)=\G. \ee Hence the claim follows from
reversibility. Indeed, put $T_-=e^{(\G-\d)\b}$. Since every path
$\o\colon\square\to\blacksquare$ has to cross $\partial
\cA_{\square}$, we can write \be{cp1}
P_{\square}(\t_{\blacksquare}\le T_-) \le P_{\square}(\t_{\partial
\cA_{\square}} \le T_-) = \sum_{t=1}^{T_-}\sum_{\xi\in \partial
\cA_{\square}} P_{\square}( \t_{\partial
\cA_{\square}}=t,\,\h_t=\xi). \ee By reversibility, we have
\be{cp2} P_{\square}( \t_{\partial \cA_{\square}}=t,\,\h_t=\xi) =
e^{-\b[H(\xi)-H(\square)]}P_{\xi}(\h_s\in \cA_{\square}\,
\forall\, 0~~\t_\blacksquare)
\leq \P_\square(\h_{\t_{\cV_{\bar n_{i_c} + 2}}}\notin\cC^*)
\le \P_\square(\t_{\cA_{>\G}}<\t_\blacksquare),
\ee
where $\cA_{>\G}= \{\h\in\cX\colon H(\h)>\G\}$. The second inequality holds
because, by Proposition \ref{sellelocali}(a), if $\h_{\t_{\cV_{\bar n_{i_c}+2}}}
\notin \cC^*$, then $\max_sH(\h_s)>H(\cC^*)=\G$. Estimate
\be{split1}
\P_\square(\t_{\cA_{>\G}}<\t_\blacksquare)
\le \P_\square(\t_\blacksquare>e^{(\G+\d)\b})
+ \P_\square(\t_{\cA_{>\G}}< e^{(\G+\d)\b}).
\ee
The first term in the right-hand side tends to zero as $\b\to\infty$ by
Proposition \ref{1}. Let $\e_0>0$ be such that $\min_{\h\in \cA_{>\Gamma}}
H(\h)\ge \G+\e_0$. Then, by using reversibility as in (\ref{cp1}--\ref{cp3}),
we may estimate the second term in the right-hand side by
\be{split1*}
\P_\square(\t_{\cA_{>\G}}\Gamma}|e^{(\G+\d)\b}e^{-(\G+\e_0)\b},
\ee
which also tends to zero as $\b\to\infty$ when we pick $0<\d<\e_0$.
\subsection{Proof of Theorem \ref{t2}}
\label{S3.3}
Let $\h\subseteq P(m_c-1,m_c-\d_c,m_c)$. Then, by Proposition
\ref{sl}(a), we have $\Phi(\h,\square)<\Phi(\h,\blacksquare)$. Fix
$0<\e_0<\Phi(\h,\blacksquare)-\Phi(\h,\square)$. Then \be{defa}
\cA_{\square}^{\Phi(\h,\square)+\e_0}
=\{\h'\in\cX\colon\,\Phi(\h',\square)<\Phi(\h,\square)+\e_0\} \ee
is a non-trivial cycle containing $\h$ and $\square$, but not
$\blacksquare$. By applying the general result on cycles in
Proposition \ref{p2.3.2}(b), we obtain the first line in
(\ref{shrgrw}). The proof of the second line in (\ref{shrgrw}) for
the case $\h\supsetneq P(m_c-1,m_c-\d_c,m_c)$ is similar.
%%%%%%%%%%%%%%% SECTION 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Additional path properties}
\label{S4}
In Section \ref{S1.5}, Item 8, we introduced two types of sets of
configurations: those denoted by $\cR$, which are given explicitly
in terms of a geometric description, and those denoted by $\cD$,
which have a more complicated definition in terms of communication
height, namely, configurations that can be reached from a
configuration of type $\cR$ by a path on a manifold $\cV_n$, for a
suitable $n$, with a maximal energy threshold $2U$. In particular,
the set of critical configurations $\cC^*$, which plays a crucial
role in the present paper, is a set of type $\cD$ (recall
(\ref{Ddef}) and (\ref{Ccrit})). Thus, in order to obtain
geometric information on $\cC^*$ we need to investigate the effect
of allowing this threshold $2U$, which results in a motion of
particles along the border of a droplet.
In Section \ref{S4.1} we describe the border motion, both for the
two- and for the three-dimensional model. In Section \ref{S4.2} we
obtain some information on the geometry of configurations in
$\cC^*$, while in Section \ref{S4.3} we offer some reflections on
the tube of typical nucleation paths.
\subsection{Motion along border of droplet}
\label{S4.1}
A particle can move away from a droplet, travel as a free particle
for awhile and then return to the droplet, but it can also move
along the border of the droplet. In fact, it can either move on a
face of the droplet, following a two-dimensional border motion, or
it can move from one face to another while staying attached to the
droplet, following a three-dimensional border motion. This border
motion is a special feature of Kawasaki dynamics and plays an
important role in determining the geometry of the critical
configurations and of the typical nucleation path.
\subsubsection{Two dimensions}
\label{S4.1.1}
Let us first consider the two-dimensional model, which was studied
in den Hollander, Olivieri and Scoppola \cite{HOS1}, \cite{HOS2}.
Here the metastable regime is $\D\in (U,2U)$. The two-dimensional
border motion is illustrated in Fig.\ 6: with the help of a free
particle, particles can slide from one side to another at maximal
energy cost $U$ (replacing the maximal energy cost $2U$ in three
dimensions).
\vglue1cm
\centerline{\epsfxsize=8cm\epsffile{trentino.eps}}
\bigskip
\botcaption{Fig.\ 6}{\,Two-dimensional motion along the border of the droplet.}
\noindent In Fig.\ 6, pictures 3--13 show the border motion
triggered by a free particle between its arrival to (pictures
1--2) and departure from (pictures 14-15) the droplet. Note that
the configurations in pictures 3, 6, 8, 10 and 13 all have the
same energy, while the energy of the configurations in pictures 4,
5, 7, 9, 11 and 12 is $U$ higher.
Due to this border motion, the gate for the nucleation and the
tube of typical nucleation paths are completely different from
those for Glauber dynamics, where the gate is given by an
$l_c\times (l_c-1)$ quasi-square with a protuberance on one of the
longest sides. The configurations given by the same quasi-square
but with the protuberance on one of the shortest sides are
dead-ends (see Ben Arous and Cerf \cite{BAC96}). In contrast, for
Kawasaki dynamics the gate for the nucleation is given by a larger
set of configurations, containing the configuration given by an
$(l_c-1) \times l_c$ quasi-square with a protuberance on one of
the longest sides plus a free particle, but also containing the
configuration given by the same quasi-square with the protuberance
on one of the shortest sides plus a free particle. Indeed, the
latter is not a dead-end, since it is easy to check that the path
obtained by completing the shortest side to obtain an
$(l_c-1)\times(l_c+1)$ rectangle, adding a protuberance on one of
the longest sides of this rectangle and sliding particles along
the border from the shortest side to the side of the protuberance
(as shown in Fig.\ 6) is made up of configurations having an
energy smaller than the initial one.
\subsubsection{Three dimensions}
\label{S4.1.2}
For the three-dimensional model studied in the present paper, the
situation is more complex. The three-dimensional border motion is
illustrated in Fig.\ 7: when a two-dimensional droplet with a
protuberance is attached to a face near the boundary of the face,
particles can slide into this face at a maximal energy cost $2U$.
\centerline{\epsfxsize=11cm\epsffile{trenino2.eps}}
\bigskip
\botcaption{Fig.\ 7}{\,Three-dimensional motion along the border
of the droplet.}
In Fig.\ 7, the configurations in pictures 1, 3, 5 and 6 all have
the same energy, while the energy of the configurations in
pictures 2, 4, 8 and 10 is $U$ higher and in pictures 7 and 9 is
$2U$ higher. Between pictures 2 and 3, particles slide one by one
along the edge of the cube. Between pictures 5 and 6, the border
motion connecting pictures 1 and 5 is repeated until one bar of
the two-dimensional droplet attached to the face has been
completed. In pictures 7, 8, 9 and 10 a bar is moved from one side
of the two-dimensional droplet to another, so as to reach a
situation similar to picture 1: a two-dimensional droplet with a
protuberance that helps to slide into the face the rest of the
particles on the edge of the cube (not depicted further). Since
picture 10 has energy $U$ higher than picture 1, this sliding can
only follow a border motion similar to the one connecting pictures
1 and 6, but cannot continue further.
In Section \ref{S4.2} we will show that the border motion cannot
really deform the critical droplet, in the sense that all the
configurations of minimal energy obtained by this border motion
have the same circumscribing parallelepiped. This is only limited
information, but a first step towards understanding the geometry
of $\cC^*$.
\subsection{Some geometry of critical configurations}
\label{S4.2}
Let $\bar n = m_c(m_c-\d_c)(m_c-1)+l_c(l_c-1)+1$. For $\bar\h\in
\cR_{l_c-1,l_c}^{2pr}(m_c-1,m_c-\d_c,m_c)$, let $ C_{\bar\h}^{2U}$
be the set of configurations $\h$ that can be reached from
$\bar\h$ by a path $\o=(\o_1,\dots,\o_k)$, $k\in\N$, in $\cV_{\bar
n}$ such that
\be{phiseq1}
\o_1=\bar\h, \quad \o_k=\h,
\quad \max_{1\leq i\leq k} H(\o_i) \leq H(\bar\h) + 2U.
\ee
{}From Theorem \ref{tAC}(a) we know that
$\bar\h\in\cF(C_{\bar\h}^{2U})$. Hence we have (recall
(\ref{Ccrit}) and (\ref{Dunione})) \be{Critalt} \cC^* =
\left(\bigcup_{\bar\h\in
\cR_{l_c-1,l_c}^{2pr}(m_c-1,m_c-\d_c,m_c)}
\cF(C_{\bar\h}^{2U})\right)^{fp}. \ee
\bp{psella} For any
$\bar\h\in\cR_{l_c-1,l_c}^{2pr}(m_c-1,m_c-\d_c,m_c)$ and any
$\h\in\cF(C_{\bar\h}^{2U})$, \be{parall} P(\bar\h)=P(\h), \ee
where $P(\bar\h),P(\h)$ are the parallelepipeds circumscribing
$\bar\h,\h$. \ep
\bpr Let $\h\in\cF(C_{\bar\h}^{2U})$ with $\h\not=\bar\h$. Then
there exists a path $\o=(\o_1,\dots,\o_k)$, $k\in\N$, in
$\cV_{\bar n}$ such that
\be{phiseq2}
\o_1=\bar\h, \quad \o_k=\h,
\quad \max_{1\leq iH(\bar\h). \ee \epr
Let us add some comments:
\begin{itemize}
\item[(1)]
The set $C_{\bar\h}^{2U}$ contains configurations (e.g.\ with a
free particle or with a 1-protuberance) for which the
circumscribing parallelepiped is different from $P(\bar\h)$.
However, Proposition \ref{psella} shows that the set
$\cF(C_{\bar\h}^{2U})$ does not.
\item[(2)]
The configuration given by an $(m_c-1)\times(m_c-\d_c)\times m_c$
quasi-cube with $l_c-1$ missing particles on one edge and with an
$l_c\times l_c$ square on one face is a configuration in
$\cF(C_{\bar\h}^{2U})$. Indeed, this configuration can be obtained
from $\bar\h$ by a three-dimensional border motion (see Fig.\ 7).
\item[(3)]
Proposition \ref{psella} fails in two dimensions: if we denote by
$\bar\h$ the configuration given by an $(l_c-1)\times l_c$
quasi-square plus a protuberance, then there are configurations
$\h\in\cF(C_{\bar\h}^{U})$ with $P(\bar\h)\not=P(\h)$ (where now
$P(\bar\h),P(\h)$ denote the rectangles circumscribing
$\bar\h,\h$). Indeed, $\h$ can be any shift of $\bar\h$ obtained
via a path that stays inside $C_{\bar\h}^{U}$. Fig.\ 8 illustrates
how, with the help of a free particle, the droplet can move
up/down/left/right as a result of a border motion. Due to
Proposition \ref{psella}, this shift is not possible in three
dimensions.
\vglue1cm \centerline{\epsfxsize=5cm\epsffile{mossa.eps}}
\bigskip
\botcaption{Fig.\ 8}{\, Upward movement of a $3 \times 3$ square.}
\noindent In Fig.\ 8, pictures 3--13 show the diffusive motion
triggered by a free particle between its arrival to (pictures
1--2) and departure from (pictures 14--15) the droplet. Note that
the configurations in pictures 3, 6, 8, 10 and 13 all have the
same energy, while the energy of the configurations in pictures 4,
5, 7, 9, 11 and 12 is $U$ higher.
\item[(4)]
Proposition \ref{psella} cannot be easily improved. There are
configurations in $\cF(C_{\bar\h}^{2U})$ for which the face of the
droplet in $\bar\h$ containing the quasi-square with the
2-protuberance looks {\it completely different}. Such
configurations can be obtained not only by completing the row, as
already noted in (2), but also by producing a completely different
shape on the face. In fact, the two-dimensional motion on the face
below energy threshold $2U$ is even richer than the one below
threshold $U$ illustrated in Fig.\ 6 and can produce {\it all\/}
two-dimensional droplets with the same area and perimeter (for
which there is a large degeneracy: e.g.\ a $5\times 4$
quasi-square with a protuberance versus a $3\times 7$ rectangle).
This degeneracy can in principle be described in full detail, but
it is only part of the problem to understand the geometry of the
set $\cC^*$. Understanding the three-dimensional border motion is
a much harder problem and is connected to the degeneracy found in
Alonso and Cerf \cite{AC96} for the isoperimetric inequalities
when the number of particles is not a magic number (see the remark
made below Theorem \ref{tAC}). Complete control of this degeneracy
seems to be a hard problem. As noted in (3), in three dimensions
the mobility of droplets is smaller than in two dimensions.
However, the mobility along the border of droplets is larger.
Thus, the two cases are rather different.
\item[(5)]
Proposition \ref{psella} is not sufficient to exclude from $\cC^*$
the configurations in which the critical two-dimensional droplet
is attached to the wrong face of the quasi-cube. We believe that
the three-dimensional motion along the border of the critical
droplet is not rapid enough to enlarge the gate as much as it does
in two dimensions, but we have no proof.
\end{itemize}
\subsection{Tube of typical nucleation paths}
\label{S4.3}
We close this paper with a heuristic discussion of the ``tube of
typical nucleation paths'', i.e., the typical behavior of the
process in the time interval
$[\theta_{\square,\blacksquare},\t_\blacksquare]$. For the case of
Glauber dynamics for the Ising model, Ben Arous and Cerf
\cite{BAC96}, Theorem 7.36, contains a complete description of
this tube. For the present case of Kawasaki dynamics for the
lattice gas model, we have only limited reflections to offer.
Because of (\ref{gate}), we can divide the nucleation time
interval in a subcritical part and a supercritical part:
\be{divisione} [\theta_{\square,\blacksquare},\t_\blacksquare]=
[\theta_{\square,\blacksquare},\tau_{\square,\cC^*,\blacksquare}]
\cup (\tau_{\square,\cC^*,\blacksquare},\t_\blacksquare]. \ee We
have some control over the subcritical part, due to our
identification in Proposition \ref{sellelocali} of the minimal
saddles between consecutive magic manifolds. However, the
supercritical part, which is relatively simple for Glauber
dynamics, is more complicated for Kawasaki dynamics.
\begin{itemize}
\item[--] {\it Supercritical\/}:
In two dimensions the supercritical growth for Kawasaki dynamics
is qualitatively different from that of Glauber dynamics. There
are arguments showing that, for Kawasaki dynamics, the
two-dimensional motion along the border of the droplet rapidly
turns a rectangle into a square or a quasi-square, while for
Glauber dynamics this mechanism is absent. Therefore, the
supercritical growth follows squares and quasi-squares for
Kawasaki, while it follows (randomly growing) rectangles for
Glauber (see Ben Arous and Cerf \cite{BAC96}). In three
dimensions, as noted in (4) in Section \ref{S4.2}, the motion
along the border of a droplet is less rapid than in two
dimensions. We therefore believe that the supercritical growth for
Kawasaki dynamics is similar to that of Glauber dynamics.
\item[--] {\it Subcritical\/}:
For the subcritical growth we can apply Olivieri and Scoppola
\cite{OS1}, Theorem 2, to study the first exit from the set
$\cA_\square$ in (\ref{Asquaredef}), i.e., the maximal connected
set of configurations containing $\square$ and having energy
$<\G$. The rough idea is the following. Look at the configurations
in $\cF(\partial\cA_\square)$ and look at the first descent from
these configurations to $\cF(\cA_\square)=\square$. The tube of
typical paths making up this first descent defines a ``standard
cascade'', consisting of a sequence of minimax's towards
$\square$, decreasing in energy and interspersed with sequences of
downhill paths and ``permanence sets'' (which are a kind of
generalized cycles). By using reversibility, we find that the
exiting tube, starting from $\square$ and ending in
$\partial\cA_\square$, can be obtained via a time-reversal
transformation from the tube describing the first descent to
$\square$. More precisely, by Theorem \ref{t1}, we know that the
minimal energy on $\partial\cA_\square$ is attained in $\cC^*$ and
that $\cA_\square$ only contains configurations with a number of
particles $~~