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\begin{document}
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%%\numberwithin{equation}{section}
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\newtheorem{corollary}{Corollary}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\title{\sc ${\mathcal L}^r$-Convergence of
a Random Schr\"odinger to a Linear
Boltzmann Evolution }
\author{Thomas Chen}
\address{Courant Institute of Mathematical Sciences\\
New York University\\
251 Mercer Street\\
New York, NY 10012-1185.}
\email{chenthom@cims.nyu.edu}
\address{New address as of September 2004: Department of
Mathematics, Princeton University, Fine Hall, Washington Road,
Princeton, NJ 08544.}
\date{}
\maketitle
\begin{abstract}
We study the macroscopic scaling and weak coupling
limit for a random Schr\"odinger
equation on $\Z^3$.
We prove that the Wigner transforms of a large class
of "macroscopic" solutions converge in
$\p$-th mean to solutions of a linear Boltzmann equation,
for any finite value of $\p\in\R_+$.
This extends previous
results where convergence in expectation was established.
\end{abstract}
\section{Introduction}
We consider the quantum mechanical dynamics of an electron
against a background
lattice of impurity ions exhibiting randomly distributed
interaction strengths. Models of this type (Anderson model) are widely used
to investigate qualitative features of technically highly relevant classes of
materials that comprise semiconductors.
Questions of key mathematical interest, treated intensively in the literature,
address the emergence of electric conduction and insulation.
While there exist landmark mathematical results explaining
disorder-induced insulation at strong coupling (Anderson localization,
\cite{aimo, frsp}), the
weak coupling regime is at present far less understood.
In the latter context, we shall here analyze issues regarding the derivation
of macroscopic transport equations.
We study the macroscopic scaling and weak coupling
limit of the quantum dynamics generated by the Hamiltonian
$$
H_\omega=-\Delta+\lambda \sum_{y\in\Z^3} \omega_y\delta(x-y)
$$
on $\ell^2(\Z^3)$. Here, $\Delta$ is the nearest neighbor discrete
Laplacian, $0<\lambda\ll1$ is a small coupling constant that defines the
disorder strength, and $\omega_y$ are independent, identically distributed
Gaussian random variables.
Let $\phi_t\in\ell^2(\Z^3)$ be the solution of the Schr\"odinger equation
\eqn
\left\{
\begin{array}{rcl}
i\partial_t\phi_t&=&H_\omega\phi_t \\
\phi_0&\in&\ell^2(\Z^3) \;,
\end{array}
\right.
\eeqn
with a non-random initial condition $\phi_0$
which is supported on a region of diameter $O(\lambda^{-2})$.
Let $W_{\phi_t}(x,v)$ denote its {\em Wigner transform}, where
$x\in\Z^3$, and $v\in[0,1]^3$.
We consider a scaling for small $\lambda$, in which
$(T,X):=\lambda^{2}(t,x)$, $V:=v$, are the
macroscopic time, position, and velocity variables (while
$(t,x,v)$ are the microscopic ones).
We then focus on a corresponding, appropriately rescaled version
$W^{resc}_\lambda(T,X,V)$ of
$W_{\phi_t}(x,v)$.
It was proved by Erd\"os and Yau for the continuum case, \cite{erdyau, erd},
and by the author for the lattice case, \cite{ch}, that for all test
functions $J(X,V)$, and globally in macroscopic time $T$,
\eqnn
&&\lim_{\lambda\rightarrow0}\Exp\Big[
\int dX dV J(X,V)W_\lambda^{resc}(T,X,V)\Big]
\nonumber\\
&&\hspace{3cm}=\int dX dV J(X,V)F(T,X,V)\;,
\eeqnn
where $F(T,X,V)$ is the solution of a linear Boltzmann equation.
The corresponding local in $T$ result was achieved much earlier
by Spohn, \cite{sp}.
Our main result in this paper improves the mode of convergence
in that we establish convergence in $r$-th mean for every finite
$r\in\R_+$,
\eqnn
&&\lim_{\lambda\rightarrow0}\Exp\Big[\,
\Big|\int dX dV J(X,V)W_\lambda^{resc}(T,X,V)
\nonumber\\
&&\hspace{3cm}-\int dX dV J(X,V)F(T,X,V)\Big|^r\,\Big]=0\;.
\eeqnn
Hence in particular, the variance of
$\int dX dV J(X,V) W_\lambda^{resc}(T,X,V)$
vanishes in this limit.
As an immediate corollary, one obtains convergence in probability,
and other standard modes of convergence.
The proof comprises generalizations and extensions to the graph expansion method
introduced by Erd\"os and Yau in \cite{erdyau, erd}, and
further elaborated on in \cite{ch}. The structure of graphs entering
the problem at hand is significantly more complicated.
But as it turns out, it is not very hard to show that
the corresponding amplitudes are extremely small. The strategy then consists
of balancing these small amplitudes against the very large number of graphs,
which is similar to the approach in the previous works \cite{erdyau, erd, ch}.
The present work, as well as \cite{ch} and \cite{erdyau, erd}, address a
time scale of order $O(\lambda^{-2})$, in which ballistic behavior
is prevalent, because the average number of collisions experienced
by the electron is finite. For this reason,
the macroscopic dynamics is governed by a Boltzmann equation.
Beyond this time scale, the
average number of collisions is {\em infinite}, and the
problem becomes much harder.
Very recently, Erd\"os, Salmhofer and Yau
have established that in
a time scale of order $O(\lambda^{-2-\kappa})$
for an explicit numerical value of $\kappa>0$, the macroscopic
dynamics in $d=3$ derived from the quantum dynamics
is determined by a {\em diffusion equation}, \cite{erdsalmyau}.
This is a breakthrough result which is the first of its kind.
We note that control of the macroscopic dynamics up to a time scale
$O(\lambda^{-2})$ allows to obtain lower bounds of a comparable scale
on the localization lengths of eigenvectors
of $H_\omega$ in 3 dimensions, for small $\lambda$, \cite{ch}.
This extends recent results of Shubin, Schlag and
Wolff, \cite{shscwo}, who derived lower bounds of this type for the
weakly disordered Anderson model in dimensions $d=1,2$ by use of
techniques of harmonic analysis.
This work comprises a partial joint result with Laszlo Erd\"os
(Lemma {~\ref{conngrbd-1}}), to whom the author is deeply grateful for his
support and generosity.
\section{Definition of the model and statement of main results}
We consider the discrete random Schr\"odinger operator
\eqn
H_\omega = -\Delta + \lambda V_\omega \;
\label{Homega-def}
\eeqn
on $\ell^2(\Z^3)$, with potential function
\eqn
V_{\omega}(\vx) =\sum_{ \vy \in\Z^3 } \omega_\vy \delta(\vx-\vy) \;,
\eeqn
where $\omega_y$ are independent, identically distributed Gaussian random variables
satisfying $\Exp[\omega_x]=0$, $\Exp[\omega_x^2]=1$, for all $x\in\Z^3$.
Expectations of higher powers of $\omega_x$ satisfy Wick's theorem,
cf. \cite{erdyau}, and our discussion below.
We use the convention
\eqn
\hat f(\vk) = \sum_{\vx\in\Z^3}
e^{-2\pi i\vk\cdot\vx} f(\vx) \; \; , \; \;
\check g(\vx)
= \int_{\Tor^3} dk\; g(\vk) e^{2 \pi i \vk\cdot\vx} \;
\eeqn
for the Fourier transform and its inverse.
In frequency space, the nearest neighbor lattice Laplacian defines
the multiplication operator
\eqn
(-\Delta f)\hat{\;}(\vk) = e(\vk) \hat f(\vk) \;,
\eeqn
where
\eqn
e(\vk) &=& 2\sum_{i=1}^3 \big( 1- \cos(2\pi k_i) \big)
\nonumber\\
&=&4\sum_{i=1}^3 \sin^2(\pi k_i)
\label{kinendef}
\eeqn
is the quantum mechanical kinetic energy of the electron at momentum $k$.
Let $\phi_t\in \ell^2(\Z^3)$ denote the solution of the random Schr\"odinger equation
\eqn
\left\{
\begin{array}{rcl}
i\partial_t\phi_t&=&H_\omega \phi_t \\
\phi_0&\in& \ell^2(\Z^3)\;,
\end{array}
\right.
\eeqn
for a fixed realization of the random potential.
Its (real, but not necessarily positive) Wigner transform
$W_{\phi_t}:\Z^3\times\Tor^3\rightarrow\R$ is defined by
\eqn
W_{\phi_t}(x,v)=\sum_{y }\overline{\phi_t(x+\frac y2)}\phi_t(x-\frac y2)
e^{2\pi iyv} \;,
\eeqn
and we note that Fourier transformation with respect to the variable $x$ yields
\eqn
\hat W_{\phi_t}(\xi,v)=\overline{\hat\phi_t(v+\frac\xi2)}\hat\phi_t(v-\frac\xi2)\;,
\label{FTWx}
\eeqn
for $\xi\in\Tor^3$.
The Wigner transform will serve as our key tool in the derivation of the
macroscopic limit for the quantum dynamics described by (~\ref{RSE}).
For this purpose,
we introduce macroscopic variables $T:=\epsilon t$, $X:= \epsilon x$, $V:=v$,
and consider the rescaled Wigner transform
\eqn
W^\e_{\phi_t}(X,V):=\e^{-3}W_{\phi_t}(\e^{-1}X,V)
\eeqn
for $T\geq0$, $X\in(\e\Z)^3$, and $V\in\Tor^3$.
For a Schwartz class function $J\in \Sc(\R^3\times \Tor^3)$, we write
\eqn
\langle J,W^\e_{\phi_t}\rangle :=
\sum_{X\in(\e\Z)^3}\int_{\Tor^3}dV \overline{J(X,V)}W^\e_{\phi_t}(X,V)
\; ,
\label{eqn3-1-0}
\eeqn
where $J_\e(x,v):=J(\e x,v)$. Let $\hat W^\e_{\phi_t}$
be defined as in (~\ref{FTWx}). Then,
\eqn
(~\ref{eqn3-1-0})&=& \langle \hat J,\hat W^\e_{\phi_t}\rangle
\nonumber\\
&=&
\int_{\Tor^3\times\Tor^3 }d\xi dv
\overline{\hat J_\e(\xi,v)}
\hat W_{\phi_t}(\xi,v) \; ,
\label{eqn3-1-1}
\eeqn
where we have defined
\eqn
\hat J_\e(\xi,v)&:=&\e^{-3}\hat J(\xi/\e,v)
\;,
\eeqn
which is a smooth delta peak of width $\e$ with respect to the $\xi$-variable,
and uniformly bounded with respect to $\e$ in the $v$-variable.
The macroscopic limit under this scaling
is determined by the linear Boltzmann equations, as was proven in \cite{ch} for
the 3-dimensional lattice model, and non-Gaussian distributed random potentials
(the Gaussian case follows also from \cite{ch}).
The corresponding result for the 2- and 3-dimensional
continuum model was proven in \cite{erdyau}.
\begin{theorem}\label{Boltzlimthm}
For $\e>0$, we consider the initial data
\eqn
\phi_0^\e(x):=\e^{\frac32} h(\e x) e^{\frac{i S(\e x)}{\e}} \;,
\label{phi0-def}
\eeqn
where $h$ and $S$ are Schwartz class functions on $\R^3$.
Let $\phi_t^\e$ be the solution of the random
Schr\"odinger equation
\eqn
\label{RSE}
i\partial_t\phi_t^\e = H_\omega \phi_t^\e
\eeqn
with initial condition given by $\phi_0^\e$, and let
\eqn
W_T^{(\e)}(X,V):=
W^{\e}_{\phi_{\e^{-1}T}^{\e} }
(X,V)
\eeqn
denote its corresponding rescaled Wigner transform.
If the scaling factor $\e$ is set equal to $\e=\lambda^2$,
where $\lambda$ is the coupling constant in (~\ref{Homega-def}),
it follows that
\eqn
\lim_{\lambda\rightarrow0}\Exp\big[
\langle J, W_T^{(\lambda^2)} \rangle\big]
=\langle J, F_T \rangle\;,
\eeqn
where $F_T(X,V)$ solves the linear Boltzmann equation
\eqn
&&\partial_T F_T(X,V) +\sum_{j=1}^3 (\sin2\pi V_j) \partial_{X_j} F_T(X,V)
\nonumber\\
&&\hspace{2cm}= \int_{\Tor^3} dU \sigma(U,V)
\lb F_T(X,U) - F_T(X,V)\rb \;.
\label{linB}
\eeqn
The collision kernel is given by
$$
\sigma(U,V):=4\pi\delta(e(U)-e(V)) \;.
$$
and the initial condition by
\eqn\label{initcondweaklim}
F_0(X,V) = |h(X)|^2\delta(V-\nabla S(X))\;,
\eeqn
which is the weak limit of $W_{\phi^\e_0}^\e$
as $\e\rightarrow0$.
\end{theorem}
The present work aims at an extension of that result
by significantly improving the mode of convergence.
Our main result is the following theorem.
\begin{theorem}\label{mainthm}
For any fixed, finite
$\p\in2\N$, $T>0$, and for any Schwartz class function $J$,
the estimate
\eqn
\Big(\Exp\Big[\Big|\bra J,W_T^{(\lambda^2)}\ket-
\Exp\big[\bra J,W_T^{(\lambda^2)}\ket\big]\Big|^\p\Big]
\Big)^{\frac{1}{\p}}\leq c(r,T)
\lambda^{\frac{1}{300\p}} \;
\eeqn
is satisfied
for $\lambda$ sufficiently small, and a constant $c(r,T)$
that does not depend on $\lambda$.
Consequently, convergence in $\p$-th mean,
\eqn
\lim_{\lambda\rightarrow0}
\Exp\Big[\Big|\bra J,W_T^{(\lambda^2)}\ket-
\bra J,F_T\ket\Big|^\p\Big]=0 \;,
\label{lpconvmainthm}
\eeqn
holds for any finite $\p,T\in\R_+$.
\end{theorem}
We remark that in particular, the variance
of $\bra J,W_T^{(\lambda^2)}\ket$ vanishes in the
limit $\lambda\rightarrow0$. We shall here not list
further standard modes of convergence
implied by (~\ref{lpconvmainthm}), which
can be found in textbooks on probability theory,
but only state the most important implication in
the following corollary.
\begin{corollary}
The rescaled Wigner transform $W_T^{(\lambda^2) }$ convergences
in probability
weakly to a solution of the linear Boltzmann equations,
globally in $T>0$, as $\lambda\rightarrow0$.
That is, for any Schwartz class function $J$, and any finite $T>0$,
\eqn
\lim_{\nu\rightarrow0}{\Bbb P}\Big[\lim_{\lambda\rightarrow0}
\Big|\bra J,W_T^{(\lambda^2) }\ket-\bra J,F_T\ket\Big|>\nu\Big]
=0 \;,
\label{convprobmainthm}
\eeqn
where $F_T$ solves (~\ref{linB}) with initial condition
(~\ref{initcondweaklim}).
\end{corollary}
\section{Proof of Theorem {~\ref{mainthm}}}
To begin with, we expand $\phi_t$ into the truncated Duhamel series
\eqn
\phi_t=\sum_{n=0}^N \phi_{n,t}+R_{N,t} \;,
\eeqn
where
\eqn
\phi_{n,t}&:=&(-i\lambda)^n \int_{\R_+^{n+1}} ds_0\cdots ds_n
\delta(\sum_{j=0}^n s_j-t)
\nonumber\\
&&\hspace{1cm}\times\,e^{i s_0 \Delta}V_\omega e^{is_1\Delta}
\cdots V_\omega e^{is_n\Delta}\phi_0
\eeqn
denotes the $n$-th Duhamel term, and where
\eqn
R_{N,t}=-i\lambda\int_0^t ds e^{-i(t-s)H_\omega}V_\omega \phi_{N,s} \;
\eeqn
is the remainder term. The number $N$ remains to be optimized.
In frequency space,
\eqn
\hat \phi_{n,t}(k_0)&=&(-i\lambda)^n\int ds_0\cdots ds_n
\delta(\sum_{j=0}^n s_j-t)
\nonumber\\
&&\times\,\int_{(\Tor^3)^n}dk_1\cdots
dk_n e^{-is_0 e(k_0)} \hat V_\omega(k_1-k_0)e^{-is_1 e(k_1)}
\cdots
\nonumber\\
&&\hspace{2.5cm}
\cdots\hat V_\omega(k_{n}-k_{n-1})
e^{-is_n e(k_n)}\hat \phi_0(k_n) \;.
\eeqn
Representing the delta distribution by an oscillatory integral, we find
\eqn
\hat \phi_{n,t}(k_0)&=&(-i\lambda)^n e^{\e t}\int_{\R} d\alpha
e^{-it\alpha}
\nonumber\\
&&\times\,\int_{(\Tor^3)^n}dk_1\cdots
dk_n \frac{1}{e(k_0)-\alpha-i\e}\hat V_\omega(k_1-k_0)
\nonumber\\
&&\hspace{2 cm}
\cdots \hat V_\omega(k_{n}-k_{n-1})\frac{1}{e(k_n)-\alpha-i\e}
\hat \phi_0(k_n) \;.
\label{hatphint-expans}
\eeqn
The function $\frac{1}{e(k)-\alpha-i\e}$ is referred to as a
{\em particle propagator}, corresponding to the frequency space representation
of the resolvent $\frac{1}{-\Delta-\alpha-i\e}$.
Likewise, we note that (~\ref{hatphint-expans}) is
equivalent to the $n$-th term in the resolvent expansion of
\eqn
\phi_t=\frac{1}{2\pi i}\int_{-i\e+\R} dz e^{-itz}\frac{1}{H_\omega-z}\phi_0
\;.
\eeqn
By analyticity of the integrand in (~\ref{hatphint-expans})
with respect to the variable $\alpha$,
and due to its decay properties as $Im(z)\rightarrow-\infty$,
the path of the $\alpha$-integration
can, for any fixed $n\in\N$, be deformed
away from $\R$ into the closed contour
\eqn
\Ie=I_0\cup I_1 \label{defIloop}
\eeqn
with
\eqnn
I_0 &:=& [-1, 13]\\
I_1&:=& ([-1, 13]-i)\cup (-1-i(0,1]) \cup (13-i(0,1]) \;,
\eeqnn
which encloses ${\rm spec}\big(- \Delta -i\e\big) = [0,12]-i\e $.
The initial condition $\phi_0\equiv\phi_0^\e$ in the random Schr\"odinger equation
(~\ref{RSE}), as characterized in (~\ref{phi0-def}), satisfies
\eqn
|\hat\phi_0(k)|=\sqrt{\delta_\e(q-k)}\;,
\eeqn
for some $q\in\Tor^3$,
where $\delta_\e$ is a smooth bump function localized in a ball of radius
$\e$, normalized by $\int_{\Tor^3} dk \, \delta_\e(k)=1$.
In particular,
\eqn
\|f\hat\phi_0\|_{L^1(\Tor^3)}&<&c\e^{\frac32}\| f\|_{L^\infty(\Tor^3)}\;,
\nonumber\\
\|\hat\phi_0\|_{L^2(\Tor^3)}&=&1\;,
\label{initcond-ass}
\eeqn
for any $f\in L^\infty(\Tor^3)$.
Applying the partial time integration method introduced in
\cite{erdyau}, we choose $\kappa\in\N$ with
$1\ll\kappa\ll\e^{-1}$, and subdivide $[0,t]$ into $\kappa$ subintervals
bounded by the equidistant points $\theta_j=\frac{jt}{\kappa}$,
where $j=1,\dots,\kappa$.
Then,
\eqn
R_{N,t}=-i\lambda\sum_{j=0}^{\kappa-1}e^{-i(t-\theta_{j+1})H_\omega}
\int_{\theta_j}^{\theta_{j+1}} ds \, e^{-isH_\omega}V_\omega \phi_{N,s} \;.
\eeqn
Let $\phi_{n,N,\theta}(s)$ denote the $n$-th Duhamel term conditioned on
the requirement that the first $N$ collisions occur
in the time interval $[0,\theta]$, and all remaining $n-N$ collisions
take place in the time interval $(\theta,s]$.
Further expanding $e^{-isH_\omega}$ into a truncated Duhamel series with
$3N$ terms, we find
\eqn
R_{N,t}=R_{N,t}^{(<4N)}+R_{N,t}^{(4N)}\;,
\eeqn
where
\eqn
R_{N,t}^{(<4N)}&=&-i\lambda\sum_{n=N+1}^{4N-1}\tilde\phi_{n,N,t} \;,
\\
\tilde\phi_{n,N,t}&:=&-i\lambda
\sum_{j=1}^{\kappa}
e^{-i(t-\theta_j)H_\omega}V_\omega\phi_{n,N,\theta_{j-1}}(\theta_{j})
\eeqn
and
\eqn
R_{N,t}^{(4N)}=-i\lambda \sum_{j=1}^{\kappa}e^{-i(t-\theta_j)H_\omega}
\int_{\theta_{j-1}}^{\theta_j}ds \;
e^{-i(\theta_j-s)H_\omega}
V_\omega \phi_{4N,N,\theta_{j-1}}(s) \;.
\eeqn
Clearly, the Schwarz inequality implies that
\eqn
\|R_{N,t}^{(<4N)}\|_2 \leq (3N\kappa) \sup_{N0$. This in turn
implies that (~\ref{mainvarest-1}) holds for any fixed,
finite $\p\in\R_+$, globally in $T$.
This proves Theorem {~\ref{mainthm}}.
\section{Graph expansions and main technical lemmata}
The key technical lemmata required to establish (~\ref{mainvarest}) are
formulated this section. The method of proof is based on graph expansions
and estimation of singular integrals in momentum space, generalizing the
analysis in \cite{erdyau} and \cite{ch} to cover the case of convergence
in $\p$-th mean.
\begin{lemma}
\label{mainlm1}
For any fixed $\p\geq2$ with $\p \in2\N$,
and $\bar n:= n_1+n_2$, where $n_1,n_2\leq N$,
\eqn
\Big(\Exptc \Big[\big|\langle \hat J_\e,\hat W_{t;n_1,n_2}\rangle
\big|^\p\Big]\Big)^{\frac1\p}\leq
\e ( (\frac{\bar n \p}{2} ) !)^{\frac{1}{\p}}
(\log\frac1\e)^{3}
(c\lambda^2\e^{-1}\log\frac1\e)^{\frac{\bar n}{2}} \;.
\label{keyest-1}
\eeqn
Furthermore, for any fixed $\p\geq2$, $\p\in2\N$, and $n\leq N$,
\eqn
\Big(\Exp\Big[ \|\phi_{n,t}\|_2^{2\p}
\Big]\Big)^{\frac1\p}\leq
((n\p)!)^{\frac{1}{\p}}(\log\frac1\e)^{3}
(c\lambda^2\e^{-1}\log\frac1\e)^{n} \;.
\label{aprioribd-1}
\eeqn
\end{lemma}
The estimate (~\ref{keyest-1}) is the key ingredient
in our analysis.
The central insight is that
for every $\p\geq 2$, the expectation over {\em 2-connected graphs} (cf. Definition
{~\ref{Expnd-def}} below)
is a factor $\e^\gamma$ smaller than the a priori bound
(~\ref{aprioribd-1}), for some $\gamma>0$.
\begin{lemma}
\label{mainlm2}
For any fixed $\p\geq2$, $\p\in2\N$, and $Nn+1}\frac{1}{e(p_{\ell'}^{(j)})-\beta_j+i\e_j}\;.
\eeqn
Clearly, it follows that
\eqn
|\amp_{\hat J_\e}(\pi)|&\leq& \lambda^{2s\bar n}
e^{2s\e t}\Big(\int_{\Tor^3} d\xi \sup_{p\in\Tor^3}
|\hat J_\e(\xi,p)|\Big)^s
\cdot (I)\cdot(II) \;,
\eeqn
where
\eqn
\int_{\Tor^3} d\xi \sup_{p\in\Tor^3}|\hat J_\e(\xi,p)|0$.
\endprf
\section{Proof of Lemma {~\ref{mainlm2}}}
Based on the previous discussion, is straightforward to see that
\eqn
&&\Exp\big[\|\phi_{n,N,\theta_{j-1}}(\theta_j)\|_2^{2r}\big]
\nonumber\\
&&\hspace{1cm}
=e^{2r\e \theta_j}\int_{(\Ie\times \bar \Ie)^r}
\prod_{j=1}^r d\alpha_jd\beta_j \,e^{-i\theta_j\sum_{j=1}^r(\alpha_j-\beta_j)}
\nonumber\\
&&\hspace{1.5cm}\times\,
\int_{(\Tor^3)^{(\bar n+2)r}}\prod_{j=1}^r d\up^{(j)}
\delta(p_n^{(j)}-p_{n+1}^{(j)})
\Exp\Big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\Big]\,
\nonumber\\
&&\hspace{1.5cm}\times
\,
\prod_{j=1}^r K^{(j)}_{n,N,\theta_{j-1},\kappa}
[\up^{(j)},\alpha_j,\beta_j,\e ]
\hat\phi_0(p_0^{(j)})\overline{\hat\phi_0(p_{\bar n+1}^{(j)})} \;.
\label{ExpphinN-1}
\eeqn
The key differences between this expression and the integrals
(~\ref{End-1}) considered above, are that now, we only study the special case
$n=n_1=n_2$, that we replace the distinguished vertex
$\hat J_\e(\xi,v)$ by $\delta(\xi)$, and that
we are now considering the full instead of the non-disconnected expectation.
We will refer to $\delta(\xi)$ as
the "{\em $L^2$-delta}", since it is responsible for the $L^2$-inner
product on the left hand side of (~\ref{ExpphinN-1}).
Furthermore,
\eqn
&&K^{(j)}_{n,N,\theta_{j-1},\kappa}
[\up^{(j)},\alpha_j,\beta_j,\e ]
\nonumber\\
&&\hspace{1cm}:=
\prod_{\ell_1=0}^{n-N}
\frac{1}{e(p_{\ell_1}^{(j)})-\alpha_j-i\kappa\e}
\prod_{\ell_2=n-N+1}^{n}
\frac{1}{e(p_{\ell_2}^{(j)})-\alpha_j-\frac{i}{\theta_{j-1}}}
\label{Kj-def-1}\\
&&\hspace{1.5cm}\times\,
\prod_{\ell_3=n+1}^{n+N}
\frac{1}{e(p_{\ell_3}^{(j)})-\beta_j+\frac{i}{\theta_{j-1}}}
\prod_{\ell_4=n+N+1}^{2n+1}
\frac{1}{e(p_{\ell_4}^{(j)})-\beta_j+i\kappa\e} \;,
\nonumber
\eeqn
cf. the discussion following (~\ref{phinNtheta-2}).
The expectation in (~\ref{ExpphinN-1}) again decomposes into a sum of
products of delta distributions, and the corresponding contributions
to (~\ref{ExpphinN-1}) can be represented by Feynman graphs.
Referring to the notational conventions introduced after (~\ref{End-1}), we
have $\bar n=2n$.
Correspondingly,
let $\Pi_{r;\bar n,n}$ denote the set of graphs on $r$ particle lines,
each containing $\bar n$ vertices from copies of
the random potential $\hat V_\omega$, and with the $L^2$-delta located
between the $n$-th and the $n+1$-st
$\hat V_\omega$-vertex. For $\pi\in\Pi_{r;n,\bar n}$, let $\amp_\delta(\pi)$
denote the amplitude corresponding to the graph $\pi$, given by the
integral obtained from replacing
$\Exp\big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\big]$ in (~\ref{ExpphinN-1}) by
$\delta_\pi(\up^{(1)},\dots,\up^{(r)})$ (the product of delta distributions
corresponding to the contraction graph $\pi$). The subscript in $\amp_\delta$
implies that instead of $\hat J_\e$ as before, we now have the $L^2$-delta
at the distinguished vertex.
Let $\Pi_{r;\bar n,n}^{conn}$ denote the subclass of $\Pi_{r;\bar n,n}$ comprising
completely connected graphs.
Then, the following estimate holds.
\begin{lemma}
\label{conngrbd-2}
Let $s\geq1$, $s\in\N$, and let
$\pi\in\Pi_{s;2n,n}^{conn}$ (that is, $\bar n=2n$) be a completely connected graph.
Then,
\eqn
|\amp_\delta(\pi)|\leq \e^{2(s-1)}(\log\frac1\e)^{s+2}(c\lambda^2\e^{-1}
\log\frac1\e)^{ sn} \;.
\eeqn
\end{lemma}
\prf
The proof is completely analogous to the one given for Lemma {~\ref{conngrbd-1}}
(using $\theta_{j-1},\frac{1}{\kappa\e}\leq\frac1\e$),
and shall not be reiterated here. All steps
taken there can be adapted to the proof
of Lemma {~\ref{conngrbd-2}} with minor modifications.
\endprf
In contrast to the situation in the context of Lemma {~\ref{conngrbd-1}},
the expectation in (~\ref{ExpphinN-1}) does not exclude completely disconnected
graphs. We recall that the sum over all amplitudes of completely disconnected graphs
on $r$ particle lines can be estimated by
\eqn
\sum_{\pi\in\Pi_{r;\bar n,n}^{disc}}|\amp_\delta(\pi)|
\leq\Big(\sum_{\pi\in\Pi_{1;\bar n,n}^{conn}}|\amp_\delta(\pi)|\Big)^r\;.
\eeqn
The bound for $s=1$ in Lemma {~\ref{conngrbd-2}}, however, is not good enough,
since there is no factor $\e^\gamma$ for any $\gamma>0$.
Using Lemma {~\ref{conngrbd-2}} for $s=1$, one will not be able to compensate
the large number $\sim 2^{nr}(n!)^r$ of disconnected graphs.
We shall hence recall another result from \cite{ch}
(the continuum version is proved in \cite{erdyau}), formulated
in the following lemma.
\begin{lemma}
Let $\bar n=2n$.
Then,
\eqn
\sum_{\pi\in\Pi_{1;\bar n,n}^{conn}}|\amp_\delta(\pi)|\leq
\frac{(c\lambda^2\e^{-1})^n}{\sqrt{n!}}
+ (n!) \e^{\frac16}(\log\frac1\e)^{3}(c\lambda^2\e^{-1}
\log\frac1\e)^{n} \;.
\label{ch-mainbd-1}
\eeqn
\end{lemma}
The term $\frac{(c\lambda^2\e^{-1})^n}{\sqrt{n!}}$ bounds the so-called
{\em ladder contribution}, while the last term
carries an additional $\e^{\frac16}$-factor
relative to the bound provided by Lemma {~\ref{conngrbd-1}} for $s=1$.
It is obtained from {\em crossing} and {\em nesting type subgraphs} that
appear in all non-ladder graphs. This issue is discussed
in much detail in \cite{ch}, and will not be recapitulated here.
Since the number of non-ladder graphs is bounded by $n!2^n$, there is a
factor $n!$.
The sum over non-disconnected graphs can be estimated by the same bound
as in Lemma {~\ref{Nondiscsum-est1}}. The result is formulated in the
following lemma.
\begin{lemma}
\label{Nondiscsum-est2}
Let $r\in2\N$, $\bar n=2n$, and $\Pi_{r;\bar n,n}^{n-d}\subset
\Pi_{r;\bar n,n}$ denote the subclass
of non-disconnected
graphs. Then,
\eqn
\sum_{\pi\in\Pi_{r;\bar n,n}^{n-d}}|\amp_{\delta}(\pi)|\leq
(r\bar n)!\e^{2}(\log\frac1\e)^{3r-2} (c\lambda^2\e^{-1}
\log\frac1\e)^{\frac{r\bar n}{2}} \;.
\eeqn
\end{lemma}
Combining (~\ref{ch-mainbd-1}) with Lemma {~\ref{Nondiscsum-est2}},
and applying the Minkowski inequality,
the statement of Lemma {~\ref{mainlm2}} follows straightforwardly.
\section{Proof of Lemma {~\ref{mainlm3}}}
We have, for $r\in2\N$, $s\in[\theta_{j-1},\theta_j]$, $n=4N$, and $\bar n=8N$,
\eqn
&&\Exp\big[\|\phi_{4N,N,\theta_{j-1}}(s)\|_2^{2r}\big]
\nonumber\\
&&\hspace{1cm}
=e^{2r\e s}\int_{(\Ie\times \bar \Ie)^r}
\prod_{j=1}^r d\alpha_jd\beta_j \,e^{-is\sum_{j=1}^r(\alpha_j-\beta_j)}
\nonumber\\
&&\hspace{1.5cm}\times\,
\int_{(\Tor^3)^{(\bar n+2)r}}\prod_{j=1}^r d\up^{(j)}
\delta(p_{4N}^{(j)}-p_{4N+1}^{(j)})
\Exp\Big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\Big]\,
\nonumber\\
&&\hspace{1.5cm}\times
\,
\prod_{j=1}^r K^{(j)}_{4N,N,\theta_{j-1},\kappa}
[\up^{(j)},\alpha_j,\beta_j,\e ]
\hat\phi_0(p_0^{(j)})\overline{\hat\phi_0(p_{8N+1}^{(j)})} \;,
\label{ExpphinN-2}
\eeqn
where again, $\e=\frac1t$. All notations are the same as in the
proof of Lemma {~\ref{mainlm2}}, in particlar, cf. (~\ref{Kj-def-1}) for
the definition of $K^{(j)}_{4N,N,\theta_{j-1},\kappa}$.
Let $\Pi_{s;8N,4N}^{conn}$ denote the subset of $\Pi_{s;8N,4N}$ of
completely connected graphs.
\begin{lemma}
\label{conngrbd-3}
Let $s\geq1$, $s\in\N$, and let
$\pi\in\Pi_{s;8N,4N}^{conn}$
be a completely connected graph.
Then,
\eqn
|\amp_\delta(\pi)|\leq \kappa^{-2rN}
\e^{2(s-1)}(\log\frac1\e)^{s+2}(c\lambda^2\e^{-1}
\log\frac1\e)^{\frac{s\bar n}{2}} \;.
\eeqn
\end{lemma}
\prf
We adapt the proof given for Lemma {~\ref{conngrbd-1}} in the following
manner. First of all, we observe that (~\ref{ExpphinN-1}) contains $r(6N+2)$
propagators with imaginary part $\pm i\kappa\e$, and $2rN$ propagators
with imaginary part $\pm \frac{i}{\theta_{j-1}}$.
In the proof of Lemma {~\ref{conngrbd-1}}, $4rN$ out of all propagators were
estimated in $L^\infty$-norm, while the remaining ones were estimated in $L^1$.
Carrying out the same arguments line by line, we shall also estimate
$4rN$ out of all propagators in (~\ref{ExpphinN-2})
in $L^\infty$. Since there are in total only $2rN$ propagators whose denominators
carry an imaginary
part $\pm\frac{i}{\theta_{j-1}}$, it follows that at least $2rN$
propagators bounded in $L^\infty$ exhibit a denominator
with an imaginary part $\pm i\kappa\e$. Correspondingly, one obtains
an improvement of the upper bound
by a factor $\kappa^{-1}$ for each of the latter, in
comparison to the $L^\infty$-bound of $\frac1\e$ used throughout the proof of
Lemma {~\ref{conngrbd-1}}. Hence, there
is in total a gain of a factor of at least $\kappa^{-2rN}$ above the
estimate derived for Lemma {~\ref{conngrbd-1}}.
\endprf
\begin{lemma}
\label{Nondiscsum-est3}
Let $r\in2\N$. Then,
\eqn
\sum_{\pi\in\Pi_{r;8N,4N}}|\amp_{\delta}(\pi)|\leq
(4rN)!\kappa^{-2rN}(\log\frac1\e)^{3r} (c\lambda^2\e^{-1}
\log\frac1\e)^{4rN} \;.
\eeqn
\end{lemma}
\prf
Given a fixed $\pi\in\Pi_{r;8N,4N}$ with $m$ connectivity components,
let us assume that $\pi$ comprises
$s_1,\dots,s_m$ particle lines, where $\sum_{l=1}^m s_l=r$.
Then,
\eqn
|\amp_{\delta}(\pi)|&\leq&\kappa^{-2N\sum_{l=1}^m s_l}
\e^{2\sum_{l=1}^m (s_l-1)}
\nonumber\\
&&\hspace{1cm}\times\,
(\log\frac1\e)^{\sum_{l=1}^m (s_l+2)}
(c\e^{-1}\lambda^2\log\frac1\e)^{\frac{\bar n\sum_{l=1}^m s_l}{2}}
\nonumber\\
&\leq&\kappa^{-2rN}
(\log\frac1\e)^{3r}
(c\e^{-1}\lambda^2\log\frac1\e)^{4rN}\;.
\eeqn
Furthermore, $\Pi_{r;8N,4N}$ contains no more than $(4rN)!2^{4rN}$ elements.
\endprf
The corresponding sum over disconnected graphs can be estimated
by the same bound as in Lemma {~\ref{Nondiscsum-est2}}.
This proves Lemma {~\ref{mainlm3}}.
\subsection*{Acknowledgements}
The author is deeply grateful to H.-T. Yau and L. Erd\"os
for their support, encouragement, advice, and generosity.
He has benefitted immensely
from numerous discussions with them, in later stages
of this work especially from conversations with L. Erd\"os.
He also thanks H.-T. Yau
for his very generous hospitality during two visits at Stanford University.
This work was supported by a Courant Instructorship, in part by a
grant of the NYU Research Challenge Fund Program, and in part by
NSF grant DMS-0407644.
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\end{document}