Content-Type: multipart/mixed; boundary="-------------1002090502382" This is a multi-part message in MIME format. ---------------1002090502382 Content-Type: application/x-tex; name="version5.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="version5.tex" \documentclass[12pt]{article} \usepackage{amsmath,amsthm,amscd,amssymb} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{graphicx} \usepackage[english]{babel} \frenchspacing \linespread{1.2} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newtheorem{prop}{Proposition}[section] \newtheorem{corr}[prop]{Corollary} \newtheorem{theo}[prop]{Theorem} \newtheorem{lem}[prop]{Lemma} \newtheorem{defi}[prop]{Definition} \newtheorem{rem}[prop]{Remark} \newcommand{\numberfield}[1]{\mathbb{#1}} \def\pn{\par\noindent} \def\ni{\noindent} \def\1{1\!{\rm l}} \newcommand{\C}{\Bbb C} \newcommand{\er}{{\Bbb R}} \newcommand{\en}{{\Bbb N}} \newcommand{\zed}{{\Bbb Z}} %\input{tcilatex} \begin{document} \title {Lifshitz tails for continuous Laplacian in the site percolation case.} \date{} \maketitle \begin{center}\bf{{W. Kirsch }} \footnote{Fakult\"at f\"ur Mathematik und Informatik FernUniversit\"at Hagen D-58084 Hagen, Germany. e-mail: werner.kirsch@fernuni-hagen.de} and \bf{{H. Najar}} \footnote{D\'epartement de Math\'ematiques Physiques I.S.M.A.I. Kairouan, Abd Assed Ibn Elfourat Kairouan 3100. Tunisie. } \end{center} \begin{abstract}In this paper we study Lifshitz tails for continuous Laplacian in a continuous site percolation situation. By this we mean that we delete a random set $\Gamma_\omega$ from $\er^d$ and consider the Dirichlet or Neumann Laplacian on $D=\er^d\setminus\Gamma_\omega$. We prove that the integrated density of states exhibits Lifshitz behavior at the bottom of the spectrum when we consider Dirichlet boundary conditions, while when we consider Neumann boundary conditions , it is bounded from below by a van Hove behavior. The Lifshitz tails are proven independently of the percolation probability, whereas for the van Hove case we need some assumption on the volume of the sets taken out as well as on the percolation probability. \end{abstract} {\small\sf 2000 Mathematics Subject Classification:15A52, 35P05, 37A30, 47F05.\\ Keywords and phrases:spectral theory, random operators, integrated density of states, Lifshitz tails, percolation, random graphs.} \section{Introduction} The present study deals with the behavior of the integrated density of states for Laplacians in a site percolation setting on the continuum. We start by defining the main object of our study, the integrated density of states (IDS). The IDS is a concept of fundamental importance in condensed matter physics. Indeed, it is considered to measure the number of energy levels per unit volume, below a given energy. For $P$, a finite-dimensional orthogonal projection, we denote by $tr(P)$, the dimension of its range. Let $P_{(-\infty, E]}$ be the spectral projection of a random Schr\"odinger operator $H_{\omega}$ and $\Lambda_{L}$ be a cube of side length $L$ around the origin. The restriction of $P_{(-\infty,E]}$ to the cube $\Lambda_{L}$ has $tr(\chi_{\Lambda_{L}}P_{(-\infty,E]})$ as a dimension range. Here for a set $A$ we denote by $\chi_{A}$, the characteristic function of the set $A$. We denote by $|E|$, the Lebesgue measure of $E$. We consider $$N(E)=\lim_{L\to \infty}\frac{1}{\mid\Lambda_L\mid}tr(\chi_{\Lambda_L}P(-\infty,E]).\label{equ1}$$ It is called the \textbf{integrated density of states} of $% H_{\omega}$ ( IDS). See \cite{KiMa} for alternative definition of the IDS. \newline The question we are interested in here concerns the behavior of $N$ at the bottom of the spectrum of $H_{\omega}$. \newline In 1964, Lifshitz \cite{Lif} argued that, for a Schr\"{o}dinger operator of the form ${H_{\omega}=-\Delta+ V_{\omega}, }$ there exists $c_{1},c_{2}>0$ such that $N(E)$ satisfies the asymptotic: $$N(E)\simeq c_1\exp({-c_2(E-E_{0})^{-\frac{d}{2}}}),\quad\text{as} \ E\searrow E_{0}. \label{eq2}$$ Here $E_0$ is the bottom of the spectrum of $H_{\omega}$. The behavior (\ref{eq2}) is known as \textbf{Lifshitz tails}. In the last thirty years, there has been vast literature, both physical and mathematical, concerning Lifshitz tails and related phenomena. We do not try to give an exhaustive account of this literature. The work of Kirsch and Metzger \cite{KM} gives a survey of such results and basic references on this subject. Below, we give results on the IDS behavior in the context of percolation. \newline There is a long history of works which consider Hamiltonians on percolation graphs. These graphs are obtained by removing sites (site percolation) or bonds (bond percolation) from a graph, for example $\zed^d$, in a random way. For example, one might remove sites in the graph $\zed^d$ independently of each other with a probability $p$ which is the same for each site, in other words there is a site at a given $i\in\zed^d$ with probability $q=1-p$. The percolation graph (consisting of the sites \textsl{not} removed) has various connected components, the so called clusters. It is known that there is a critical value $p_c$ which depends on the dimension $d$, such that for $qp_c$ (the percolation regime) there is also an infinite cluster. For $d\geq 2$ the critical value satisfies $0p_c$ $\tilde{N}(E)$ has a van Hove behavior near $E_0=0$, i.e. $\tilde{N}(E)\simeq C E^{\frac{d}{2}}$. In this paper, we consider quantum percolation problems on $\er^d$. In one particular model we remove a set $S_i=S+i$ near the point $i\in\zed^d$ with probability $p$. More precisely, let $\{\xi_i\}_{i\in\zed^d}$ be a collection of independent $\{0,1\}$-valued random variable with $P(\xi=1)=p$. We set $D=\er^d\setminus\bigcup_{\xi_i=1} S_i$ and denote by $H_D^N$ and $H_D^D$ the Laplacian restricted to $D$ with Neumann and Dirichlet boundary conditions respectively. As the set $D$ is random, the operators $H_D^N$ and $H_D^D$ are random operators. We note that the set $D$ may be a connected set for all $\omega$ if the set $S$ is small, e.g. if $S\subset \Lambda_a = \{x\in\er^d\mid -a\leq x_i\leq a \text{ for } i=1\ldots d\}$ with $a<1$. On the other hand, if $S$ is big and $p$ is small, the set $D$ is a union of bounded connected components. We consider the behavior of the integrated density of states for these two classes of operators. We will obtain Lifshitz tails for the Dirichlet case under mild assumptions. For the Neumann case we prove that $N(E)\geq C E^{\frac{d}{2}}$ if the set $S$ is not too big. The result on the Neumann case is based on a deterministic result (Theorem \ref{th3.5}) which we believe is of some interest on its own. We apply it to other percolation models as well, for example to a Poisson percolation model. \section{Model and results} Let us first define the model we are studying. \subsection{The Model}\label{sec2.1} Consider the probability space $\Omega =\{0,1\}^{\mathbb{Z}^d}$, which is endowed with the usual product sigma-algebra, generated by finite dimensional cylinder sets, and equipped with a product probability measure $\mathbb{P}$. Elementary events in $\Omega$ are sequences $\omega \equiv (\omega_{\gamma})_{\gamma\in \mathbb{Z}^d}$, we assume their entries to be independently and identically distributed according to a Bernoulli law. i.e $$\mathbb{P}\{\omega_0=1\}=p.$$ Here $p$ is the {\textit{site }} probability, a parameter in $]0,1[$.\newline Let $\mathbb{Z}^d$ be the lattice, considered as a subset of $\mathbb{R}^d$. We set $\Lambda_L$ the hypercube defined by $\Lambda_{0}=[-\frac{L}{2},\frac{L}{2}[^d$. Let $\mathcal{S}$ be a bounded subset of $\mathbb{R}$. \newline We set $$\Gamma_{\omega}=\bigcup_{\gamma \in \mathbb{Z}^d }\omega_{\gamma}\Big(\gamma+\mathcal{S}\Big)=\bigcup_{\omega_\gamma=0} \Big(\gamma+\mathcal{S}\Big)$$ where the notation $0\;\mathcal{S}=\emptyset$ and $1\; \mathcal{S}=\mathcal{S}$ was used. Let $D^{\omega}=\mathbb{R}^d\backslash \Gamma_{\omega}$, be the domain obtained from $\mathbb{R}^d$ by taking off the random set $\Gamma_\omega$. We denote by $H_{\omega}^D$ the restriction of the Laplacian $-\Delta$ to $D^{\omega}$, with Dirichlet boundary condition and by $H_{\omega}^N$ the restriction of $-\Delta$ to $D_\omega$ with Neumann boundary condition. $H_{\omega}^D$ and $H_{\omega}^N$ are self-adjoint linear operators on $L^2(\mathbb{R}^d)$, we call them {\bf{ Dirichlet }} respectively {\bf{Neumann }} {\bf{Laplacian for site-percolation in } } $\mathbb{R}^d$. We denote by $N_D$, respectively by $N_N$ the IDS of $H^D_\omega$ respectively of $H^N_\omega$. \begin{rem} As in \cite{Cha}, we can write $H_{\omega}^D$ formally, in the Anderson form with some random potential $V_\omega$ i.e $$H_{\omega}^D=-\Delta +V_{\omega},$$ $$V_{\omega}=\sum_{\gamma\in \mathbb{Z}^d}\omega_{\gamma}f(x-\gamma). \ \ \text{with}\ \ f=\infty, {\text{on \ }}\mathcal{S} \ {\text{and \ }} 0 \ {\text{elsewhere }}.$$ \end{rem} Let us consider the map $\mathcal{P}$, from $\Omega$, to the set of the self-adjoint operator on $L^2(\mathbb{R}^d)$, such that to $\omega$ associate $\displaystyle \mathcal{P}(\omega)=H_{\omega}= H_{\omega}^{\bullet}=-\Delta^{\bullet}_{\lceil \mathbb{R}^d\backslash \Gamma_{\omega}}, \bullet\in\{N,D\}$. $\mathcal{P}$ is measurable. Sometimes we suppress the superscript $D$ and respectively $N$ and write only $H_\omega$. In this cases the statement remains true for both $H^D_\omega$ and $H^N_\omega$.\newline Let $\mathcal{T}_{i}$ be the unitary translation operator on $L^2(\mathbb{Z}^d)$, i.e $$\mathcal{T}_{i}\psi (x)=\psi (x-i),\ \ \ \forall \psi \in L^2(\mathbb{R}^d) \ \text{and}\ \ x\in\mathbb{R}^d.$$ As the probability measure $\mathbb{P}$ is ergodic with respect to the group of translation $(\mathcal{T}_{i})_{i\in \mathbb{Z}^d}$, acting as $\displaystyle \mathcal{T}_{i}(\omega)= (\omega_{\gamma+i})_{\gamma \in \mathbb{Z}^d}$, we get $$\mathcal{T}_{i}^{-1}H_{\omega}\mathcal{T}_{i}=H_{(\mathcal{T}_i\omega)},\ \ \forall i\in \mathbb{Z}^d, \ \omega\in \Omega.$$ By this, we deduce that $H_{\omega}$ is a measurable family of self-adjoint and ergodic operators. According to \cite{Kirmar, kir1,PaFi} we know that there exists $\Sigma, \Sigma _ {pp}, \Sigma _ {ac}$ and $\Sigma _ {sc}$ closed and non-random sets of ${\Bbb{R}}$ such that $\Sigma$ is the spectrum of $H_{\omega}$ with probability one and such that if $\sigma_{pp}$ (respectively $\sigma_{ac}$ and $\sigma_{sc}$) denote the pure point spectrum (respectively the absolutely continuous and singular continuous spectrum) of $H_{\omega}$, then $\Sigma _ {pp}=\sigma _ {pp}, \Sigma _ {ac}=\sigma _ {ac}$ and $\Sigma _ {sc}=\sigma _ {sc}$ with probability one.\newline The following Lemma gives the precise location of the spectrum. \begin{lem}\label{lem1} The spectrum $\Sigma$, of $H_{\omega}$ is $[0,+\infty[$ with probability one. \end{lem} {\bf \textit{Proof:}} First let us notice that for any $\omega \in \Omega$, we have $$\label{equ2.9} H_\omega \geq 0.$$ This gives that $$\Sigma \subset [0,+\infty[.$$ So one needs to show the opposite inclusion, i.e $$[0,+\infty[ \subset \Sigma\ \text{for}\ \mathbb{P}-{\text{almost\ every\ }} \omega \in \Omega. \label{equ7}$$ For this, let $\widetilde{\Omega}$, be the following events \begin{multline} \widetilde{\Omega}=\Big\{ \omega \in \Omega: \ \ \text{for any}\ k\in \mathbb{N}, \\ \text{There exists}\ \Lambda_{k}^{\omega}\subset \mathbb{R}^d, \text{such that}\ D^{\omega}_{\Lambda_{k}^{(\omega)}}=\mathbb{R}^d_{\Lambda_{k}^{(\omega)}}\Big\}.\label{equ8} \end{multline} Here $A_{\Lambda_k^{(\omega)}}$ is the set of points which are both in $A$ and $\Lambda_k^{\omega}$. In (\ref{equ8}) we asked that all sites inside of ${\Lambda_{k}^{(\omega)}}$ to be present. Let $E\in [0,+\infty[=\Sigma(-\Delta)$ be arbitrarily fixed. Using Weyl criterion, we know that there exists a Weyl sequence $(\varphi_{E,n})_{n\in \mathbb{N}}\subset L^2(\mathbb{R}^d)$ , for $-\Delta$. Thus $\|\varphi_{E,n}\|=1,$ for all $n\in \mathbb{N}$ and $$\lim_{n\to \infty} \|(\Delta+E\cdot \mathbb{I})\varphi_{E,n}\|=0$$ Notice that for any $i\in \mathbb{Z}^d$, $(\mathcal{T}_i\varphi_{E,n})_{n\in \mathbb{N}}$ is also a Weyl sequence. Without loss of generality, we assume that the sequence $(\varphi_{E,n})_{n\in \mathbb{N}}$ is compactly supported. So for any $\omega \in \widetilde{\Omega}$, there exists a Weyl sequences $(\varphi_{E,n}^{\omega})_{n\in \mathbb{N}}$ for $(-\Delta)$ on $\mathbb{R}^d$ with the property that all the supports are contained inside the cubes of (\ref{equ8}). So for every $\omega \in \widetilde{\Omega}$ and any $n\in \mathbb{N}$ there exists an integer $k_n^{\omega}$ and a cube $\Lambda_{k_n^{\omega}}^{(\omega)}$ and $\varphi_{E,n}^{\omega}$ as in (\ref{equ8}) such that $\text{supp}(\varphi_{E,n}^{\omega } )$ is contained in $\Lambda_{k_n^{\omega}}^{(\omega)}$. That is, $$\min\{\mid x-y \mid : x\in \text{supp}\varphi_{E,n}^{\omega};\ y\in \mathbb{R}^d\backslash \Lambda_{k_n^{\omega}}^{(\omega)} \} >0 .$$ So, for any $n\in \mathbb{N}$ and $\omega \in \widetilde{\Omega}$, we get $$\|(H_{\omega}-E\mathbb{I})\varphi_{E,n}^{\omega})\|=\|(\Delta+E\cdot \mathbb{I})\varphi_{E,n}^{\omega}\|.$$ Hence, $(\varphi_{E,n}^{\omega})_{n\in \mathbb{N}}$ is also a Weyl sequence for $H_{\omega}$. So we get (\ref{equ7}) for any $\omega \in \widetilde{\Omega}$. Now it suffices to check that $\mathbb{P}(\widetilde{\Omega})=1$. For this let $\lambda$ be an integer bigger than 2. $(\Lambda_{k,\lambda})_{\lambda \in \mathbb{N}}\subset \mathbb{R}^d$ be a sequence of disjoint cubes in $\mathbb{R}^d$ i.e $\Lambda_{k,\lambda_1} \cap \Lambda_{k,\lambda_2}=\emptyset$ whenever $\lambda_1 \neq \lambda_2$. We set $\Omega_{k,\lambda}=\{\omega \in \Omega:D^{\omega}_{\Lambda_{k,\lambda}}=\mathbb{R}^d_{\Lambda_{k,\lambda}}\}$ . So $(\Omega_{k,\lambda})_{\lambda \in \mathbb{N}}$, is a sequence of two by two statistically independent sets, with non-zero probability and independent of $\lambda\in \mathbb{N}$. So, Using the Borel-Cantelli lemma, we get that $\mathbb{P}(\Omega_k)=1$ for any $k\in \mathbb{N}+2$, where $$\mathbb{P}(\Omega_k)=\limsup_{\lambda \to +\infty}\Omega_{k,\lambda}.$$ The proof of Lemma \ref{lem1} is ended by noting that $$\cap_{k\in\mathbb{N}+2}\Omega_k \subset \widetilde{\Omega}.$$ $\hfill \Box$ \subsection{Main results} Our study is on the bottom of the almost sure spectrum of $H_{\omega}$. We recall that $| \mathcal{S}|$ denotes the volume of $\mathcal{S}$ \begin{theo}\label{theo1} For $p\in ]0,1[$, we have $$\lim _{E\to 0^+}\frac{\log|\log N_D(E)|}{\log E}=-\frac{d}{2}.$$ \end{theo} \begin{rem} We notice that the set $D_\omega$ may have unbounded connected component depending on the shape of the set $\mathcal{S}$ and on the probability $p$. Indeed the result of the above theorem is independent on the existence or nonexistence of such unbounded clusters. \end{rem} \begin{theo}\label{theo1bis} If $\displaystyle |\mathcal{S}|\cdot p^{\frac{1}{d}}<\frac{1}{2^d}$ is a bounded set in $\er^d$, we get $$\lim_{E\to 0^+}\frac{\log N_{N}(E)}{\log(E)}\leq \frac{d}{2}.\label{equ2.14}$$ \end{theo} \begin{rem} \begin{itemize} \item In fact we prove a slightly better result then (\ref{equ2.14}). Indeed we prove that $$N(E)\geq C\cdot E^{\frac{d}{2}},$$ with $C$ a constant which can be computed from our proof below. \item In the generality of Theorem \ref{theo1bis}, we can't bound $N(E)$ from above, as we discuss at the end of the paper. \item The result given in Theorem \ref{theo1bis} is deduced from a more general result of Theorem \ref{th3.5} which gives a deterministic lower bound of the numbers of eigenvalues for Neumann operators. \end{itemize} \end{rem} \section{The proof of Theorem \ref{theo1}:} In this section, we prove Theorem \ref{theo1}. \subsection{Preliminary}\label{sec3} We start by recalling the following result and giving some properties of the IDS. \begin{prop} Let $\varphi \in C_{0}^{\infty}(\mathbb{R}^d)$, then $$\lim_{L\to \infty}\frac{1}{|\Lambda_L|} tr(\varphi(H_{\omega})\chi_{\Lambda_L})=\mathbb{E}\Big(tr(\chi_{\Lambda_1} \varphi(H_{\omega})\chi_{\Lambda_1})\Big),\label{equ11}$$ for $\mathbb{P}$-almost all $\omega$. Here $\mathbb{E}$, is the expectation with respect to the probability measure $\mathbb{P}$. \end{prop} {\textit{\bf{Proof:}}} First we write $\chi_{\Lambda_L}=\sum_{i\in \Lambda_{L}\cap \mathbb{Z}^d}\chi_{\Lambda_1(i)}$, Here $\Lambda_1(i)$, is the cube of center $i$ end side length $1$. We set $\zeta_i=tr(\varphi(H_{\omega})\chi_{\Lambda_1(i)}).$ So $\zeta_i$ is an ergodic sequence (with respect to $\mathbb{Z}^d$) of random variables. So $$\frac{1}{|\Lambda_L|} tr(\varphi(H_{\omega})\chi_{\Lambda_L})=\frac{1}{|\Lambda_L|}\sum_{i\in \Lambda_{L}\cap \mathbb{Z}^d}\zeta_i. \label{equ12}$$ By the Birkhoff's ergodic theorem, the sum in (\ref{equ12}) converges to its expectation value. This ends the proof of (\ref{equ11}).$\hfill\Box$\newline Now, we notice that both sides of (\ref{equ11}), are positive linear functionals on the bounded, continuous functions. So, they define positives measures $\mu_{L}$ and $\mu$ respectively, i.e$$\int_{\mathbb{R}}\varphi(\lambda)d\mu_{L} (\lambda)=\frac{1}{|\Lambda_L|} tr(\varphi(H_{\omega})\chi_{\Lambda_L})$$ and $$\int_{\mathbb{R}}\varphi(\lambda)d\mu (\lambda)=\mathbb{E}\Big(tr(\chi_{\Lambda_1} \varphi(H_{\omega})\chi_{\Lambda_1})\Big).$$ For those two measures we have the following result proved in \cite{Ves1}, \begin{theo}For any $\varphi\in C_{0}^{\infty}({\Bbb{R}})$ and for almost all $\omega\in \Omega$ we have $$\lim_{k\rightarrow \infty }\langle \varphi,d\mu_{L}\rangle=\langle \varphi,d\mu\rangle.$$ \end{theo} \begin{rem} We call the non-random probability measure $\mu$ {\textit{the density of states measure}}. It satisfies the following fundamental properties $$N(E)=\mu((-\infty,E]),$$ $$\Sigma(H_{\omega})={\textrm{supp}(\mu)}.$$ \end{rem} Let $H_{\Lambda_L}^D(\omega)$ be the operator $H_{\omega}^D(\omega)$ restricted to $\Lambda_L$ with Dirichlet boundary condition also on $\partial \Lambda_L$ , while $H_{\Lambda_L}^N(\omega)$ when we consider Neumann boundary condition on $\partial \Lambda_L$. As $H_{\omega}$ is an elliptic operator, $H_{\Lambda_L}^{\bullet}(\omega),\ \bullet\in \{D,N\}$, has a compact resolvent . So $H_{\Lambda_L}^{\bullet}(\omega)$ has a discrete spectrum with possibility of accumulation at the infinity \cite{ReSi}. Let us denote the sequences of eigenvalues by $$E_1^{\bullet}(\Lambda_L)\leq E_2^{\bullet}(\Lambda_L)\leq \cdots \leq E_n^{\bullet}(\Lambda_L)\leq \cdots$$ For $E\in \mathbb{R}$ we denote by $N(H_{\Lambda_L}^\bullet (\omega),E)$ the number of eigenvalues of $H_{\Lambda_L}^\bullet (\omega)$ less or equal to $E$, of course counted with their multiplicities. \newline To prove Theorem \ref{theo1}, we prove a lower and an upper bounds on $N(E)$. The upper and lower bounds are proven separately and based on the following result (Theorem 5.25 p. 110 of \cite{PaFi}). $$\frac{1}{\mid\Lambda_L \mid}\mathbb{E}\{N(H_{\Lambda_L}^D(\omega),E)\} \leq N(E)\leq \frac{1}{\mid\Lambda_L \mid} \mathbb{E}\{N(H_{\Lambda_L}^N(\omega),E)\}. \label{exe1}$$ Inequalities in (\ref{exe1}) are based on Akcoglu-Krengel subergodic theorem. Indeed $$H_{\Lambda_1}^N(\omega)\oplus H_{\Lambda_2}^N\leq H_{\Lambda_1 \cup \Lambda_2}^N (\omega)$$ and $$H_{\Lambda_1 \cup \Lambda_2}^D(\omega) \leq H_{\Lambda_1}^D(\omega)\oplus H_{\Lambda_2}^D(\omega),$$ hold on $L^2(\Lambda_1 \cup \Lambda_2),$ for all bounded cubes $\Lambda_1,\Lambda_2\subset \mathbb{R}^d$ whenever $\Lambda_1 \cap \Lambda_2=\emptyset$ \cite{ReSi}. \subsection{The Dirichlet case:} \subsubsection{The upper bound} The upper bound is proved by a comparison procedure. Indeed, let, $$\widetilde{V}_{\omega}=\sum_{\gamma \in \mathbb{Z}^d}\omega_{\gamma} \chi_{\mathcal{S}}(x-\gamma),$$ and $$\widetilde{H}_{\omega}=-\Delta +\widetilde{V}_{\omega}$$ For any $\Lambda_L\subset \mathbb{R}^d$, we set $$\mathcal{Q}^{\Lambda_L}_{\omega}(\varphi,\psi)=\langle\varphi, H_0 \psi\rangle,\ \ \ \varphi,\psi \in H_0^1(\Lambda_L\backslash \Gamma_{\omega})=\mathcal{D}_{\Lambda_L},$$ and $${\widetilde{\mathcal{Q}}^{\Lambda_L}}_{\omega}(\varphi,\psi)=\langle\varphi, \widetilde{H}_{\omega} \psi\rangle,\ \ \ \varphi,\psi \in H_0^1(\Lambda_L)=\widetilde{\mathcal{D}}_{\Lambda_L}.$$ From \cite{ReSi}, we recall the following result, \begin{lem}\label{ha1} For any $L,k\in \mathbb{N}^*$, we have \begin{multline}\sup_{\varphi_1,\cdots, \varphi_{k-1}\in \widetilde{\mathcal{D}}_{\Lambda_L}}\inf_{\psi\in [\varphi_1,\cdots,\varphi_{k-1}]^{\bot}\cap {\widetilde{\mathcal{D}}}_{\Lambda_L} , \|\psi\|=1}{\widetilde{\mathcal{Q}}}^{\Lambda_L}_{\omega}(\psi,\psi)\leq \\ \sup_{\psi_1,\cdots,\psi_{k-1}\in\mathcal{D}_{\Lambda_L}}\inf_{\varphi\in [\psi_1,\cdots,\psi_{k-1}]^{\bot}\cap\mathcal{D}_{\Lambda_L} ,\|\varphi\|=1}\mathcal{Q}^{\Lambda_L}_{\omega}(\varphi,\varphi) . \end{multline} \end{lem} From Lemma \ref{ha1}, one deduces that for any $n\in \mathbb{N}^*$, we have $$E_n(\widetilde{H}_{\omega}(\Lambda_L))\leq E_n(H_{\omega}(\Lambda_L)) .$$ Thus, we get that for any $E\in \mathbb{R}$, $$N(H_{\Lambda_L}(\omega),E)\leq N(\widetilde{H}_{\Lambda_L}(\omega),E)$$ We notice that for $\widetilde{H}_{\omega}$ it is already known that it exhibits Lifshitz tails, by the result of Kirsch and Simon \cite{KirSim2}. This ends the proof of the upper bound. $\hfill \Box$ \subsubsection{The lower bound} Let $H_1^D$ be the operator $H_\omega^D$ with $\omega_\gamma=1$ for any $\gamma \in \mathbb{Z}^d$. We recall that for any $\Lambda_L\subset \mathbb{R}^d$, we have $$H_\omega \leq H_{\Lambda_L}^D(\omega) \leq H_{1,\Lambda_L}^D.$$ So by the min-max argument we get that $$E_{1}^D(\Lambda_L)\leq E_{1}^D(H_{1,\Lambda_L}).$$ Using equation (\ref{exe1}) one gets, \begin{eqnarray} N(E)&\geq&\frac{1}{L^d}\cdot \mathbb{P}\{E_1^D(\Lambda_L)\leq E\}\nonumber \\ &\geq & \frac{1}{L^d}\cdot \mathbb{P}\{E_{1}^D(\Lambda_{L}) \leq E \ \text{and}\ \forall \gamma\in \Lambda_L\cap \mathbb{Z}^d,\ \omega_\gamma=0\} \\ &\geq &\frac{1}{L^d}\cdot \mathbb{P}\{E_1^D(H_{0,\Lambda_L})\leq E \ \text{and}\ \forall \gamma\in \Lambda_L\cap \mathbb{Z}^d,\ \omega_\gamma=0\}.\label{ver4} \end{eqnarray} If $L$ is such that $$E_1^D(H_{0,\Lambda_L})\leq E , \label{equl}$$ then $$(\ref{ver4}) \geq \frac{1}{L^d}\cdot \mathbb{P}\{\omega_0=0\}^{|\Lambda_L|}=(1-p)^{L^{d}}\label{equ}.$$ We remark that (\ref{equl}) is satisfied for $L\thickapprox E^{-\frac{1}{2}}$. This ends the proof of the lower bound. \section{The proof of Theorem \ref{theo1bis}} \subsection{The deterministic results} The aim of this section is to prove the following theorem \pagebreak[3] \begin{theo}\label{th3.5} Let $\Omega$ be a subset of $\mathbb{R}^d$. Let $H^N$ be the self adjoint operator defined as the Laplacian operator, $-\Delta$ restricted to $L^2(\mathbb{R}^d\backslash \Omega)$ with Neumann boundary conditions on $\partial \Omega$. Assume there exists a sequence of cubes $\Lambda_L$ such that for $L$ big enough we have $\displaystyle\frac{2^d}{L^d}\mid\Omega_L\mid <1$ ($\displaystyle\Omega_L=\Lambda_L\cap\Omega$). Then, then there is a constant $C$, such that for $E>0$ small enough the IDS $N$ of $H^N$ satisfies $$\lim_{E\to 0^+}\frac{\log N_N(E)}{\log E}\leq \frac{d}{2}.$$ \end{theo} {\bf{Proof:}} For the proof of we start by recalling the following Lemma from \cite{kirwarz}: \begin{lem}\label{kir} Let $n\in \mathbb{N}$ and $\varphi_1,\cdots ,\varphi_n\in \mathcal{H}$ be in the domain $\mathcal{D}$ of a self-adjoint operator $A$, which acts on a separable Hilbert space $\mathcal{H}$. Suppose there are constants $\alpha_1\leq \cdots \leq \alpha_n\leq \alpha$ such that $$|\langle \varphi_i,\varphi_j\rangle -\delta_{i,j}|\leq \varepsilon_1 \ \ \text{and}\ |\langle \varphi_i,A\varphi_j\rangle -\alpha_j\delta_{i,j}|\leq \varepsilon_2 \label{equkir}$$ for all $i,j=1,\cdots ,n$. If $\varepsilon_1<1$, then $$N(A,\frac{\alpha+\varepsilon_2}{1-\varepsilon_1})\geq n.$$ \end{lem} Before giving the proof of Lemma \ref{kir} let us use it to end the proof of Theorem \ref{th3.5}. Let us denote by $H^{ND}_{\Lambda_L}$, the operator $H^N$ restricted to $\Lambda_L$, with Dirichlet boundary condition on $\partial \Lambda_L$. $H^{ND}_{\Lambda_L}$ is the self adjoint operator associated with the following quadratic form $$\mathcal{H}_{L}^{ND}(u,v)=\int_{\Lambda_L\backslash\Omega_L}\nabla u(x)\cdot \overline{\nabla u(x)} dx;$$ with domain $$\displaystyle \mathcal{D}(\mathcal{H}_{L}^{ND})=\{ f\in H^{1}(\Lambda_L\backslash\Omega_L);\ f\lceil \partial \Lambda_L=0\}.$$ $\mathcal{H}_{L}^{ND}$ is a densely-defined quadratic form. We denote by $H^{ND}_{\Lambda_L}$ the self-adjoint operator associated to $\mathcal{H}_{L}^{ND}$.\newline For $n\in \mathbb{N}^d$, we set$$\displaystyle\alpha_n(L)=(\frac{\pi}{L})^2\sum_{i=1}^{d}n_i^2,$$ and $$\Phi_{n,L}(x)=(\frac{2}{L})^{d/2}\prod_{i=1}^{d}\varphi_{n_i}(\frac{x_i}{L}),$$ where $$\varphi_k=\cos (k\pi x); k=1,3,5,\cdots, .$$ We note that the family $(\Phi_{n,L})_{n\in \mathbb{N}^d;}$ lies in the domain $\displaystyle \mathcal{D}(\mathcal{H}^{ND})$ and that $$|\Phi_{n,L}(x)|^2~\leq ~\frac{2^d}{L^d}\ .$$ We have $$\mathcal{H}_{L}^{ND}(\Phi_{n,L},\Phi_{n,L})=\int_{\Lambda_L}|\nabla \Phi_{n,L}(x)|^2dx-\int_{\Omega_L}|\nabla \Phi_{n,L}(x)|^2dx.\label{equ33}$$ As $(\Phi_{n,L})_{n\in\mathbb{N}^d}$ are the eigenfunctions of the Laplacian restricted to the box $\Lambda_L$ with the appropriate boundary condition \cite{ReSi}, we get the following properties, \begin{itemize} \item for any $n,m\in\mathbb{N}^d$ such that $n\neq m$, we have $$\int_{\Lambda_L} \Phi_{n,L}(x)\cdot \overline{\Phi_{m,L}(x)} dx=\int_{\Lambda_L} \nabla\Phi_{n,L}(x)\cdot \overline{\nabla \Phi_{m,L}(x)} dx=0,\label{equ34}$$ \item for any $n\in \mathbb{N}^d$, $$\int_{\Lambda_L}|\nabla \Phi_{n,L}(x)|^2 dx=\alpha_{n}(L).\label{equ35}$$ \end{itemize} Now using equation (\ref{equ33}) and taking into account (\ref{equ34}) and (\ref{equ35}) we get that, $$|\mathcal{H}_{L}^{ND}(\Phi_{n,L},\Phi_{n,L})-\alpha_n(L)|\leq \alpha_n(L)\cdot (\frac{2}{L})^d|\Omega_L|$$ and $$| \mathcal{H}_{L}^{ND}(\Phi_{n,L},\Phi_{m,L})|\leq (\alpha_n(L))^{\frac{1}{2}}(\alpha_m(L))^{\frac{1}{2}}\cdot (\frac{2}{L})^d\cdot |\Omega_L|.$$ Applying Lemma \ref{kir}, with $$\alpha=E>0,\ \varepsilon_1=(\frac{2}{L})^d\cdot \mid \Omega_L\mid <1,\ \varepsilon_2=\alpha \cdot (\frac{2}{L})^d\cdot \mid \Omega_L\mid ,$$ we get that there exist constants $C_1,C_2>0$ such that $$N(H^{ND}_{\Lambda_L},C_1\cdot E)\geq C_2\cdot E^{\frac{d}{2}}\cdot L^d.\label{equ38}$$ The right hand side of equation (\ref{equ38}) is due to the fact that for $-\Delta$, there exists $C_2>0$ such that \cite{ReSi} $$\sharp\{n; \alpha_{n}(L)\leq E\}=C_2\cdot E^{d/2}L^d.$$The proof of Theorem \ref{th3.5} is ended by taking into account (\ref{equ38}) and the following equation \cite{kir1} $$\frac{1}{L^d}\cdot N(H_{\Lambda_{L}}^{ND},E)\leq N(E).$$ \hfill $\square$ \newline {\bf{The proof of Lemma \ref{kir}:}} We start by recalling the following expression of the counting function given in terms of a supremum of the dimension of all linear subspace $\mathcal{E}$ in the domain of $A$ which we denote by $\mathcal{D}$. $$N(A,E)=\sup_{\mathcal{E}\subset \mathcal{D}}\{dim \mathcal{E}| \langle \varphi,A\varphi\rangle \leq E\langle \varphi, \varphi \rangle , \text{for\ all }\varphi \in \mathcal{E}\}.$$ When $\varepsilon_1<1$, $\displaystyle \varphi_1,\cdots, \varphi_n$ span a subspace $\mathcal{E}_n$ of dimension $n$. For any $\Phi \in \mathcal{E}_n$ there exist (non-unique) coefficients $c_1,\cdots, c_n\in \mathbb{C}$ such that $$\Phi=\sum_{i=1}^nc_i\varphi_i.$$ Using (\ref{equkir}), one gets $$\langle \Phi,\Phi\rangle \geq \sum_{i=1}^n|c_i|^2-\sum_{i,j}^n|c_i|\cdot |c_j|\cdot |\langle \varphi_i,\varphi_j \rangle-\delta_{i,j}|\geq (1-\varepsilon_1)\sum_{i=1}^n|c_i|^2,$$ and $$\langle \Phi,A\Phi\rangle \leq \sum_{i=1}^n\alpha_i|c_i|^2+\sum_{i,j=1}^n|c_i|\cdot |c_k|\cdot |\langle \varphi_i,A\varphi_j,\rangle -\alpha_i\delta_{i,j}|\leq (\alpha+\varepsilon_2)\sum_{i=1}^n|c_i|^2.$$ The proof of Lemma \ref{kir} is ended by setting $\displaystyle E=\frac{\alpha+\varepsilon_2}{1-\varepsilon_1}$. \subsection{Examples}Let us consider some particular cases. \subsubsection{The periodic case.} Consider $$\Omega=\bigcup _{\gamma\in \mathbb{Z}^d}\Lambda_{\beta}(\gamma),\beta<1,$$ here $\Lambda_\beta(\gamma)$ is the cube of center $\gamma$ and side length $\beta$. In this particular case we get $|\Omega_L|=L^d\beta^d$, and the assumption of Theorem \ref{th3.5} is satisfied for $\beta<\frac{1}{2}$. \begin{rem} The periodic model includes random displacement models. \end{rem} \subsubsection{The Anderson case.} Consider now the case of the model described in subsection \ref{sec2.1}, which is the object of our studies. $$\Omega=\bigcup_{\gamma\in\mathbb{Z}^d}\omega_\gamma(\mathcal{S}+\gamma).$$ In this situation we get: For $P$-almost all $\omega$ and any $\alpha>\mid \mathcal{S}\mid \cdot p$ and all $L$ large enough we have $$\frac{1}{L^d}|\Omega_L|\leq \alpha\label{equr}$$ So the assumption of Theorem \ref{th3.5} is satisfied for $\mid\mathcal{S}\mid \cdot p<\frac{1}{2^d}$. To get (\ref{equr}), we use the fact that $\mathcal{S}$ is bounded, so there exists $L_0$ such that $\mathcal{S}\subset \Lambda_{L_0}$, and we get $$\Omega_L=\cup_{\gamma \in \mathbb{Z}^d}\omega_\gamma(\mathcal{S}+\gamma)\cap \Lambda_L\subset \cup_{\gamma \in \Lambda_{L_0+L}}\omega_\gamma(\mathcal{S}+\gamma).$$ So \begin{eqnarray} \frac{1}{L^d}\cdot |\Omega_L| &\leq & \frac{1}{L^d}\sum_{\gamma \in \Lambda_{L_0+L}}\omega_\gamma \cdot |\mathcal{S}|\\ &\leq & \frac{(L_0+L)^d}{L^d}\cdot |\mathcal{S}|\cdot (\frac{1}{(L_0+L)^d}\sum_{\gamma \in \Lambda_{L_0+L}}\omega_\gamma). \label{rect} \end{eqnarray} Now by taking $L$ big enough we get the result as the term between parenthisis in eqution (\ref{rect}), converges to $\mathbb{E}(\omega_0)=p$ by the strong law of large numbers. \subsubsection{The Poisson case.} Let us starts by giving the model. Let $m(dx)$ be a Poisson random measure on $\mathbb{R}^d$. This means that if $(A_1, \cdots, A_n)$ are pairwise disjoint Borel sets in $\mathbb{R}^d$, the random variables $m(A_1),\cdots m(A_n)$ are independent, and when $A$ is a bounded Borel set, for $k=0,1, \cdots,$ the random variable has the distribution $$\mathbb{P}\{m(A) =k\}=e^{-c|A|}\frac{(c|A|)^k}{k!}.$$ Here $c$ is the the concentration defined by $\mathbb{E}(m(A))=c\cdot |A|.$ If $(\xi_i)_i$ are the atoms of the Poisson measure then $m$ can be written as $$m(dx)=\sum_i\delta_\xi(x).$$\newline In the above notation we have $$\Gamma_\omega=\Omega =\bigcup_i\,(S+\xi_i).$$ So we get that $$\Omega_L\subset \bigcup_{\xi_i\in \Lambda_{L_0+L}}(\mathcal{S}+\xi_i).$$ In the same way as in the Anderson Model we get that \begin{eqnarray} \frac{1}{L^d}\cdot |\Omega_L| &\leq & \frac{1}{L^d}\cdot \cap_{\xi_i\in \Lambda_{L_0+L}}|(\mathcal{S}+\xi_i)|\nonumber \\ &\leq & \frac{(L_0+L)^d}{L^d}\cdot |\mathcal{S}|\cdot (\frac{1}{(L_0+L)^d} \cdot m(\Lambda_{L_0+L}). \end{eqnarray} As $$\frac{1}{|\Lambda_k|}m(\Lambda_k)\longrightarrow c \ \ {\text{if}}\ \ k\longrightarrow +\infty.$$ we get that the assumption of Theorem \ref{th3.5} is satisfied for $$c\cdot |\mathcal{S}|<\frac{1}{2^d}.$$ $\mathbf{Acknowledgements.}$ \textit{W. K. would like to thank the I.S.M.A.I. Kairouan for the warm hospitality extended to him there.} \textit{H. N. gratefully acknowledges the financial support of the Institute f\"ur Mathematik and SFB/TR 12 of the Deutsche Forschungsgemeinschaft. He also thanks the Ruhr-Universit\"at Bochum and the FernUniversit\"at Hagen for the warm hospitality extended to him.} \begin{thebibliography}{Tototo} \smallskip \bibitem[1]{Bar} M.T Barlow: {\sl \textit{Random walks on supercritical percolation clusters.}} Ann. Prob. {\bf{32}} p. 1061-1101, (1995). \bibitem[2]{Cha} J. T. Chayes, L. Chayes, J. R. Franz, J. P. Sethna and S.A. Trugman: {\sl \textit{On the density of states for the quantum percolation problem.}} J. Phy. A: Math. Gen. {\bf{19}} p. 1173-1177, (1986). \bibitem[3]{Gen1} P-G. de Gennes, P. Lafore and J. 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