Thermodynamic Limit for Interacting Particles in Random Media

# The Existence of the Thermodynamic Limit for the System of Interacting Quantum Particles in Random Media

## Abstract.

The thermodynamic limit of the internal energy and the entropy of the system of quantum interacting particles in random medium is shown to exist under the crucial requirements of stability and temperedness of interactions. The energy turns out to be proportional to the number of particles and/or volume of the system in the thermodynamic limit. The obtained results require very general assumptions on the random one-particle model. The methods are mainly based on subadditive type inequalities.

The author is partially supported by the grant ANR-08-BLAN-0261-01.

## 1. Introduction

Since the fundamental work [And58] of P. W. Anderson, the theory of random Schrödinger operators has been an extensively studied field of mathematical physics. The greatest attention has been paid since now to the one-particle approximation and we do not try to list here even the major works on this topic.

There are relatively few papers where finitely many particles are considered. There is a series of papers by Michael Aizenman and Simone Warzel that generalize the techniques of fractional moments method (see, for example, [AW09]) and another series of articles by Victor Chulaevsky, Yuri Suhov and their collaborators that make use of the multiscale analysis [CS09, CBdMS11]. The common point of these works is that they consider the number of particles being fixed and study the infinite volume limit for such system.

The present paper is an attempt to give an insight of what happens if both the number of particles and the volume go to infinity together so that the number of particles per unit volume is kept constant.

The same question has already been addressed by various authors in case of absence of background potential, i.e., when one-particle propagation is given by pure Laplacian. In this paper, we will frequently follow the framework developed by David Ruelle in [Rue99], though the presence of random potential presents certain mathematical difficulty, which we will explain later. We would also like to refer to an outstanding article [LL72] of Elliot H. Lieb and J. L. Lebowitz, where Coulomb interactions (always in absence of background potential) are treated.

The idea that the number of particles grows with the volume looks natural in the context of condensed matter physics. As a reference real-world example consider a piece of metal or semiconductor. A bigger piece should contain proportionally more electrons. As macroscopic objects are composed of many atoms (Avogadro constant ), and thus, ions and electrons, it turns out that the corresponding mathematical notion is the thermodynamic limit. Its existence for thermodynamic quantities, such as internal energy, free energy, calorific capacity, and so on, is the mathematical verification of the fact that these quantities are extensive. The latter is barely assumed in physics but actually needs rigorous verification.

Let us briefly discuss the mathematical objects we study. All the notions will be introduced later in full regularity. Let

 Hω=−\laplaced+Vω (1.1)

be the random Schrödinger operator that describes a single quantum particle in random environment . Kinetic part is -dimensional Laplacian. One may also consider magnetic Schrödinger operator or whatever, provided that a number of basic facts, such as Wegner estimate (see Proposition 5.3), from the theory of one particle random operators hold true. Actually, the whole ideology of this paper is that we take one-particle operators for known and deduce on this base properties for multiparticle operators.

The restriction of to the domain is denoted by . For one particle Hamiltonian as in (1.1), we define, with a slight abuse of notation, the particle operator (restricted in physical space to domain ) with pair interactions potential by

 Hω(Λ,n)=−\laplacend+n∑i=1Vω(xi)+∑i≠jU(xi−xj),

where , , are particles’ coordinates.

Using the notations introduced above, the general question we want to understand is the behavior of in the thermodynamic limit:

 Hω(Λ,n)→?,|Λ|→∞,n→∞,n/|Λ|→\const. (1.2)

In this paper, we answer a much more modest question than (1.2). Namely, let be the ground state energy of . In Theorem 3, we show, in particular, that the ground state energy per particle admits the thermodynamic limit:

 ∃lim\frEω(Λ,n)n,|Λ|→∞,n→∞,n/|Λ|→\const. (1.3)

Moreover, the same theorem gives a bit more general result that allows to scale on the eigenenergy number in the spectrum. Roughly, the eigenenergy number (counting function) should be of order of exponent of the number of particles to ensure the convergence.

Theorem 3 gives the reciprocal result interchanging roles of energy and the counting function in the spectrum (the theorem is stated in terms of entropy which is the logarithm of counting function).

The main tool we use to obtain our results is a modified version of subadditive ergodic theorem (see Proposition 4.1). For instance, one may show that the ground state energy is additive with respect to the pair up to an error term that can be taken into account. To make use of subadditivity we follow the construction of D. Ruelle [Rue99]. Nevertheless, significant modifications are made in the proof because of the fact that instead of full translation invariance of free Laplacian, we have only the covariance property of the family of random operators. In general, we are only able to prove the convergence in with respect to randomness (see Theorem 3 case (1)). A stronger convergence in and almost surely is established for compactly supported interactions (Theorem 3 case (2)).

In the last part of the present paper, we consider the system of noninteracting fermions in random medium. We show that nontrivial effects arise due to Fermi-Dirac statistics even in absence of interactions. In particular, we give an exact expression for the limit (1.3) in terms of the one particle density of states measure (see Theorem 5.3) and we find an interesting relation with the Fermi energy.

The rest of the paper is organized as follows. The model of interacting quantum particles in random media and the notion of thermodynamic limit are introduced in Section 2. The results (mainly on the existence of thermodynamic limit) constitute Section 3, followed by the proofs in Section 4. In addition, Section 4 uncovers some extra properties of the energy density (see Subsections 4.3 and 4.4). The proofs themselves may be instructive as well. In Section 5, simple calculations concerning the thermodynamic limit for vanishing interactions are provided.

The author is grateful to his thesis advisor, Prof. Frédéric Klopp, for proposing the problem, for his constant interest and support, as well as many valuable discussions.

## 2. Model and Notations

### 2.1. Model of Interacting Quantum Particles in Random Media

We consider a system of interacting quantum particles in a random medium. The discrete and continuum cases are treated simultaneously and an explicit indication is given if a result is valid only for one setting. In the discrete case, the configuration space is given by and for the continuous case by . In uniform manner, the one-particle Hilbert space is given by

 \frH=\frH1=L2(\calV).

The -particle Hilbert space definition depends on the statistics (physical nature of quantum particles). The following statistics are considered.

1. The Maxwell - Boltzmann statistics. The particles are physically distinguishable and no restrictions are imposed on a multiparticle wavefunctions. This model is suitable, in particular, for the description of heavy atomic nuclei, i. e., for particles that exhibit classical properties. The corresponding Hilbert space is given by

 \frHn=n⨂j=1\frH=L2(\calVn).
2. The Bose - Einstein statistics: the particles are bosons. The wavefunction is necessarily symmetric with respect to the permutations of coordinates:

 \frHn+=\Symn\frH=L2+(\calVn),

where is the symmetrised tensor product.

3. The Fermi - Dirac statistics: they describe fermions. Wavefunctions are restricted to the antisymmetric subspace

 \frHn−=n⋀j=1\frH=L2−(\calVn),

where is the external product.

and are proper subspaces of . For , we write to denote the orthogonal projector on , where stands for the Maxwell - Boltzmann statistics, for the Bose - Einstein statistics and for the Fermi - Dirac statistics. Obviously, is the trivial projector.

One particle Hamiltonian is given by

 Hω=Hω(1)=−\laplace+Vω,

and acts on , where

• is either discrete or continuous Laplacian,

• a random potential is (at least) -ergodic and satisfies a decorrelation (independence at a distance) condition:

 ∃R0>0, such that if \dist(A,B)>R0,then {Vω(x)}x∈A and {Vω(x)}x∈B are independent.
{remark}

We also take into account the classes of random potentials that have the ergodic group reacher than -translations. For instance, everything what follows remains true for the Poisson model.

{notation}

We write , and for the associated probability space, probability measure and expectation respectively. For , we denote by the corresponding translations (measure preserving transformations) in and by the corresponding unitary transformations (coordinate shifts) in . Namely,

 Hτγ(ω)=T∗γHωTγ, (2.1)

where , , .

By we denote a corresponding operator in that acts only on the -th particle. More precisely,

 H(i)ω=\ds1\frH⊗…⊗\ds1\frHi−1 times⊗Hω⊗\ds1\frH⊗…⊗\ds1\frHn−i times. (2.2)

The -particle Hamiltonian in random environment and with interactions is given by the following self-adjoint operator on :

 Hω,♯(n)=P♯[n∑i=1H(i)ω+Wn]. (2.3)

For each , is an interaction potential given by a function of the particles coordinates , . We refer to the whole collection as interactions in general. Remark also that in this model interactions are deterministic and all particles live in the same random background potential .

In (2.3), the free part

 H0ω,♯(n)=n∑i=1H(i)ω

is called the second quantization of in context of the Fock space (see, for example, [BR97]). Namely, we have to restrict the second quantization of to the -particle subspace of the whole Fock space:

 H0ω,♯(n)=dΓ(Hω)∣∣\frHn♯,

where denotes second quantization procedure.

{remark}

acts from into itself for any choice of , whereas an arbitrary interaction potential does not necessarily preserve complete (anti)symmetry. That is why the projector a-priori acts non trivially in this formula. However, potentials that we consider later are permutation symmetric (confer Section 3, property (PI)), so that the projector becomes obsolete in (2.3), i.e.,

 Hω,♯(n)=H0ω,♯(n)+Wn.

The Dirichlet and Neumann restrictions of to a finite box are denoted by , where . is a self-adjoint operator on . We omit in notations frequently.

The operator has a discrete spectrum. We call the counting function associated to this operator

 \calNω,♯(E,Λ,n)=\card{Ek(Λ,n,ω;♯)≤E},

where are the eigenvalues of . For the reasons that will become apparent later, the entropy is a more convenient quantity:

 Sω,♯(E,Λ,n)=log\calNω,♯(E,Λ,n). (2.4)
{notation}

Sometimes we will drop some (if not all) of the indices and arguments of the counting function and the entropy. For example, if we are interested in the dependence on energy, we will write just:

 S(E)=Sω,♯(E,Λ,n),\calN(E)=\calNω,♯(E,Λ,n).
{remark}

As the counting function takes its values in , the entropy takes its values in .

{observation}

For fixed , , and , the entropy is a non-decreasing right-continuous step function.

The monotonicity of allows to define a (quasi-)inverse function . As is not a local bijection at any point, the inverse function doesn’t exist in a canonical manner. Our choice of the inverse is the following.

For such that we define

 Eω,♯(Λ,n,S)=EexpS(Λ,n,ω;♯). (2.5)

The application is a right inverse of the entropy (2.4) in the following meaning. For one has

 Sω,♯(Eω,♯(Λ,n,S),Λ,n)=S. (2.6)

Reciprocally, if , then

 Eω,♯(Λ,n,Sω,♯(E,Λ,n))=E−, (2.7)

where is the closest from below to eigenenergy of .

The relations (2.6) and (2.7) motivate this choice of an inverse function.

{definition}

We denote by the ground state energy of the operator :

 \frEω,♯(Λ,n)=infφ∈\Dom(Hω,♯(Λ,n))φ≠0⟨Hω,♯(Λ,n)φ,φ⟩∥φ∥2.

Two characterizations of the ground state energy in terms of entropy are given below.

{proposition}

is the ground state energy if and only if and or, equivalently, if and only if and .

{proposition}

Alternatively, the ground state energy is given by the zero entropy:

 \frEω,♯(Λ,n)=Eω,♯(Λ,n,0).

The latter characterization is essentially due to our choice of the inverse function given by (2.5) and would not be valid for another choice of the inverse, whereas the Proposition 2.1 is universal with respect to the particular choice of the function .

### 2.2. Thermodynamic Limit

In this section we discuss the notion of thermodynamic limit, following the approach of [Rue99]. For sake of completeness and the ease of reading, we repeat here the basic definitions related to the notion of thermodynamic limit that can be found in various monographs and articles such as [Rue99, LSSY05, Gri65, LL72].

First of all, we give a precise meaning to the notion of a sequence of domains tending to infinity.

{definition}

Let be the diameter of and be the -neighborhood of , i.e.,

 ∂hΛ=∂Λ+B(0,h),

where is the open ball of center and radius .

{definition}

The sets tend to infinity in the sense of Fisher if

 lim|Λ|=+∞

and there exists a “shape function” such that

 limα→0π(α)=0

and for sufficiently small and all

 |∂α\diam(Λ)Λ|/|Λ|≤π(α).

In what follows, we will always assume that in the sense of Fisher.

{remark}

Consider a sequence of rectangular domains. The fact that they tend to infinity in the sense of Fisher is equivalent to say that all their sides tend to infinity at a comparable speed, i.e.,

 d∏j=1[0,Lj]→∞⇔{[]lminjLj→∞,1≥minjLj/maxjLj>1/C.
{definition}

The limit , , where is a positive constant (density of particles), is called the thermodynamic limit.

Usually one is interested in extensive quantities per particle or per unit of volume (that is the same thing up to a multiplicative constant due to Definition 2.2) while considering the thermodynamic limit.

{definition}

Let be a random variable that depends on a domain , a number of particles and on a set of parameters . We say that admits the thermodynamic limit if the limit

 limΛ→∞n/|Λ|→ρ\frL[\calP]Xω(Λ,n,\calP)n

exists in some sense with respect to randomness (almost sure, in probability, in ). Here is a certain limiting procedure for the parameters , i.e., it determines the way how the parameters evolve when and go to infinity in the thermodynamic limit. For example, see (3.3), where an extra parameter is entropy , and the limiting procedure for the entropy reads as it should tend to infinity linearly with the number of particles and/or the volume of the system.

In thermodynamics, some commonly used quantities (such as internal energy, for example) are assumed to be extensive, i.e., additive with respect to volume. The existence of the thermodynamic limit is the mathematically rigorous way of verifying the above assumption. Thus, it is one of the fundamental questions of statistical physics. Some authors go even further and refer to the question of existence of thermodynamic limit purely as “existence of thermodynamics” [LL72].

In what follows, we will be primarily concerned with the existence of the thermodynamic limit for the energy with and, in particular, the ground state energy , i.e., for .

## 3. Main Results

Throughout this section we work with Dirichlet boundary conditions

 Hω(Λ,n)=HDω(Λ,n)

and we omit the explicit indication in notations. We give a series of statements concerning the existence of the thermodynamic limit for the model of interacting quantum particles in random media, which was introduced in Section 2.1. Basic properties of the thus defined limits are discussed.

We shall need some assumptions on the model that we introduce now.

Pair translation invariant interactions. The interactions are by pairs and are invariant under translations if for all

 Wn(\bfx)=∑1≤i

where is a function on . We also assume that pair interactions are symmetric: , .

Tempered interactions. Assume (PI) and that there exist , and such that for all

 |U(x)|≤A|x|−λ.

This condition (together with an additional assumption that is integrable in a neighborhood of zero) guarantees that interactions are of short range, i.e.,

 ∫\bbRd|U(x)|\rmdx<+∞.

The temperedness or similar conditions on the behavior of the interactions at the infinity have been used by various authors such as Léon van Hove, Joel L. Lebowitz, Robert B. Griffits and, in particular, Michael E. Fisher and David Ruelle. The reader is referred to [Fis64], [Rue99], [FR66], [Leb76], [Gri65].

{remark}

The above assumption of temperedness of interactions can be physically motivated by the following argument. Consider electrons in metal or semiconductor as a reference system. Though electrons interact via Coulomb potential () in vacuum, the situation is different in metal where each electron is surrounded by a “cloud” of other electrons and lives in a grid of ions. This leads to what is called screening of Coulomb potential in metal (see [AM76, Zag98]) and results to the effective interaction potential of the form

 U(r)=Qrexp(−r/λ). (3.1)

The interaction is between quasiparticles “electron+cloud”, that are called plasmons.1

Lower-bounded one particle Hamiltonian. The one-particle random operator is bounded from below uniformly with respect to randomness :

 ∃C>0, such that Hω≥−C for all ω∈Ω.
{notation}

We write . For an index set we write

 xI=(xi1,…,xin)∈\bbRnd

for the vector of the coordinates of the particles enumerated by , where the elements are ordered in a nondecreasing fashion: if .

{definition}

Let , , be a partition of particles in two disjoint subsets. The term of interaction between the particles and is given by

 WI1,I2(x\bbNn1+n2)=Wn1+n2(x\bbNn1+n2)−Wn1(xI1)−Wn2(xI2).

Repulsive interactions. The interactions are repulsive, if for all , as in Definition 3 it holds

 WI1,I2≥0.

If one assumes (Rep) and that there are no self-interactions: , then for all

 Wn(x\bbNn)≥Wn−1(x\bbNn−1)+W1(xn)≥Wn−1(x\bbNn−1)≥…≥0,x\bbNn∈\bbRnd. (3.2)

If one also assumes (PI), then (Rep) is equivalent to say that

 U≥0.

Stable interactions. The interactions are stable if there exists , such that for all

 Wn(\bfx)≥−nB.

By (3.2), repulsive interactions are stable with . The stability of interactions for various models is widely discussed, in particular, in [FR66].

Compactly supported interactions. Using the notations of Definition 3, the interactions have compact support if there exists such that

 WI1,I2(x\bbNn1+n2)=0

for all such that .

{remark}

Obviously, for pair interactions, compact support is stronger than temperedness, i.e.,

 (PI)+(Comp)⟹(PTI) with A=0.

Let us now discuss the physical validity of the above assumptions. For more details on classical electrodynamics, see, for example [Jac75] and for the electrodynamics of continuous media, see, for example [LL60].

• The model of pair translation invariant (PI) repulsive (Rep) interactions is natural for a description of identical quantum particles such as electrons.

• The condition of temperedness (PTI) might seem more restrictive at first glance, but is usually circumvented as described in Remark 3 by replacing actual interactions by screened interactions and bare electrons by quasiparticles.

• The condition of compactly supported interactions (Comp) is a technical one and allows us to treat interaction of higher order than pair (triple, etc.). However, even short range Yukawa interactions (3.1) are not compactly supported.

• The repulsive nature of interactions between identical particles (Rep) is widely accepted. Though, mathematically only the condition of stability (SI) is needed. Further discussion of stability condition and examples of catastrophic, i.e., not stable, potentials may be found in [Rue99].

• Finally, the lower boundedness of the one-particle operator (LB) seems a natural basic assumption.

The following theorem is the main result of this paper on the existence of thermodynamics for the model described in Section 2.

{theorem}

[existence of thermodynamic limit] Suppose that the one particle operator is lower bounded (LB) and that the interactions are stable (SI). Let also any of the following two cases hold:

1. interactions are translation invariant and by pairs, i.e., they satisfy (PTI)

2. interactions are compactly supported, i.e., they satisfy (Comp).

Then, the energy per particle admits thermodynamic limit, namely

 Eω(Λ,n,S)n→\calE(ρ,σ)as Λ→∞,n|Λ|→ρ,Sn→σ, (3.3)

where and . The convergence takes place in in case (1) and in and -almost sure in case (2). The limiting energy density is defined by (3.3), is a non-random function (does not depend on ) and the limit is the same if both conditions (1) and (2) are satisfied.

The energy density has the following basic properties.

{proposition}

[critical density of particles] There exists a critical density such that

 {\calE(ρ,σ)<+∞,if ρ<ρc,\calE(ρ,σ)=+∞,if ρ>ρc,

for all .

{proposition}

[energy density properties] The energy density is

1. a convex function of variables ;

2. a nondecreasing function of and ;

3. a continuous function in the region .

{corollary}

The energy density admits an inverse . The latter is convex upwards with respect to and is nondecreasing in for any fixed . p

Next we state a reciprocal result exchanging the roles of energy and entropy (the proof follows [Gri65]).

{theorem}

[existence of thermodynamic limit for entropy] Let the conditions of Theorem 3 be satisfied. Then for and

 Sω(E,Λ,n)n→σ(ρ,\calE)as Λ→∞,n|Λ|→ρ,En→\calE.

The convergence takes place in the same sense as given by Theorem 3.

{remark}

The condition that the energy belongs to the image of the function is crucial. One might remark as well that due to monotonicity and convexity properties of , either identically, or .

## 4. Proofs

This section is mainly devoted to the proof of -convergence (case (1) of Theorem 3). The basic ideas were inspired by [Rue99] and [Gri65], though the crucial difference is that instead of translation invariance of one particle operator (which is free Laplacian for both of the above works) we have ergodicity, i.e., covariance with respect to a family of measure preserving transformations of the probability space.

We assume (LB), (PTI) and (SI) throughout this section, except for Subsection 4.6, where different assumptions will be made. We also recall that the Dirichlet boundary conditions are used, i.e., .

Subadditive inequalities play the key role in our proofs. The basic idea for all the proofs for existence theorems in this paper (and many others: see, for example, [Rue99, LL72, Gri65]) may be summarized as:

• find a subadditive type inequality,

• use the existing or prove an analog of subadditive ergodic theorem that guarantees the convergence.

Next is the core lemma that gives the subadditivity of energy.

{lemma}

[Test function construction] Suppose (IAD) and (PTI) are satisfied. Let the statistics be fixed. Consider domains , such that and functions , , with energies below :

 (4.1)

Then, using , , one can construct explicitly , a function of particles defined of a unified box with energy below :

 ⟨Hω,♯(Λ1∪Λ2,n1+n2)ζ,ζ⟩≤(E1+E2+An1n2r−λ)∥ζ∥2.
{remark}

The construction of a test function is explicit in the proof that follows.

###### Proof of the Lemma.

We consider the extensions of the functions , , by zero on , which we also denote by . Remark that (4.1) implicitly contains the fact that are zeros on the respective domains boundaries (due to Dirichlet condition) so that the zero extension is a natural operation. These extensions obviously preserve (anti)symmetry when . Consequently, one has for any initial choice of .

We study each statistics separately now.

Boltzmann statistics:

Take

 ζ=φ1⊗φ2.

Then, using (PTI)

 (4.2)

where is the potential of interaction between particles in and particles in (see Definition 3). In the last inequality, we used the temperedness of pair interaction potential . As in each of terms, , and , then by (PTI)

 ∫Λn11∫Λn22\rmd\bfxU(xi−xj)|ζ(\bfx)|2≤Ar−λ∫Λn11∫Λn22\rmd\bfx|ζ(\bfx)|2≤Ar−λ∥ζ∥2,

where we used the shorthand notation for the vector of all particles’ coordinates.

Bosons:

We construct

 ζ(x\bbNn1+n2)=∑I⊂\bbNn1+n2\cardI=n1φ1(xI)φ2(x\bbNn1+n2∖I). (4.3)

This function is symmetric with respect to coordinate permutations. Moreover, the terms in the sum (4.3) are mutually orthogonal in ; hence,

 ∥ζ∥2=(n1+n2n1)∥φ1∥2∥φ2∥2,

and the operator preserves this orthogonality:

 Hω,+(Λ1∪Λ2,n1+n2)⋅φ1(xI)φ2(x\bbNn1+n2∖I)⊥φ1(xJ)φ2(x\bbNn1+n2∖J),I≠J.

Consequently, rewriting (4.2) for given by (4.3) and using the orthogonality, one obtains:

 ⟨Hω,+(Λ1+Λ2,n1+n2)ζ,ζ⟩ ≤(n1+n2n1)⋅(E1+E2+An1n2r−λ)⋅∥φ1∥2∥φ2∥2 =(E1+E2+An1n2r−λ)∥ζ∥2.
Fermions:

The construction for fermions is similar to that for bosons. We define

 ζ(x\bbNn1+n2)=∑I⊂\bbNn1+n2\cardI=n1(−1)∑i∈Ii⋅φ1(xI)φ2(x\bbNn1+n2∖I). (4.4)

This function is antisymmetric with respect to coordinate permutations. The remaining part of the proof follows exactly that for the bosons.

{remark}

The construction (4.4) is a generalization of the Slater determinant [Gre07].

{remark}

The Dirichlet boundary conditions are crucial for the proof as they provide a zero cost (canonical) extension of functions from to a larger domain without changing the norm. At this moment, we are not able to prove an analog of Theorem 3 (essentially, we need an analog of Lemma 4.1) for Neumann or periodic boundary conditions.

From now on, we omit the statistics sign in the notations. Everything that follows is valid for all the statistics. However, one should be warned that quantities (such as limiting values) may depend on the statistics.

{proposition}

Let the interactions be tempered (PTI). If , then

 Nω(Λ1∪Λ2,n1+n2,E1+E2+An1n2r−λ) ≥Nω(Λ1,n1,E1)Nω,\eps(Λ2,n2,E2), (4.5) Sω(Λ1∪Λ2,n1+n2,E1+E2+An1n2r−λ) ≥Sω(Λ1,n1,E1)+Sω(Λ2,n2,E2). (4.6)
###### Proof.

The proof of (4.5) is done using the variational principle for eigenvalues of and the function from Lemma 4.1 as a test function. Taking the logarithm, one obtains (4.6). ∎

{proposition}

Let the interactions be tempered (PTI).

1. Take , such that , . If , then

 Eω(Λ1∪Λ2,n1+n2,S1+S2)≤Eω(Λ1,n1,S1)+Eω(Λ2,n2,S2)+An1n2r−λ. (4.7)
2. Take such that , , and domains at mutual distances greater than . Then

 Eω(m⋃i=1Λi,m∑i=1ni,m∑i=1Si)≤m∑i=1Eω(Λi,ni,Si)+A2(m∑i=1ni)2r−λ. (4.8)
###### Proof.

The inequality (4.8) is an immediate consequence of (4.7). The latter is obtained by taking of (4.6) and using (2.6). ∎

### 4.2. L2-convergence on a Special Sequence of Cubes

In this section, we will construct a special sequence of cubes in configuration space , on which the existence of thermodynamic limit will be proven. The idea is inspired by [Rue99].

Let be a number that satisfies

 1<2d/λ<θ<2

and let

 \wtL>R=R0+δ2−θ, (4.9)

where is a constant that will be fixed later. For an integer put

 LN=2[12(2N\wtL−θNR)], (4.10)

so that , and define the cube by

 ΛN=[−LN/2,LN/2]d⊂\calV.

Remark that the vertices of are at integer points. According to (4.10) it is possible to place translates of (cubes ) inside at mutual distances at least

 RN=LN+1−2LN=θN(2−θ)R+\eps=θN(R0+δ)+\eps≥R0,

where is the error due to the rounding procedure. The constant is chosen to compensate a possibly negative error term , so that the last inequality holds true. It suffices, for example, to choose .

We remark that cubes are explicitly given by

 Λ(i)N=ΛN+γ(N)i,

where

 γ(N)i=LN+1−LN2⋅ei∈\bbZd,ei=(±1,…,±1)∈\bbRd,i=1,…,2d.
{remark}

It is important that the translation vectors are integer, because it ensures that the restrictions of the random potential to for different are connected by the covariance relation (2.1).

The function satisfies the following monotonicity properties.

{lemma}

For fixed and , the energy is

1. a nondecreasing function of ,

2. a nonincreasing function of .

By Lemma 4.2 and the almost-subadditivity condition (4.8) we obtain for in that

 Eω⎛⎝ΛN+1,2d∑i=1ni,2d∑i=1Si⎞⎠ ≤Eω⎛⎝2d⋃i=1Λ(i)N,2d∑i=1ni,2d∑i=1Si⎞⎠ ≤2d∑i=1Eω(Λ(i)N,ni,Si)+A2⎛⎝2d∑i=1ni⎞⎠2R−λN =2d∑i=1Eτγ(N)i(ω)(ΛN,ni,Si)+A2⎛⎝2d∑i=1ni⎞⎠2R−λN,

where is the family of ergodic transformations of , that were introduced in Notation 2.1. In particular, for

 Eω(ΛN+1,2dn,2dS)≤2d∑i=1Eτγ(N)i(ω)(ΛN,n,S)+A2(2dn)2R−λN. (4.11)

Let now and be positive numbers such that and are integer, for a sufficiently large integer . Plug in (4.11)

 nN=2Ndρ\wtLd,SN=σnN=2Ndρσ\wtLd (4.12)

for . Remark that

 nN/|ΛN|→ρ,SN/nN=σ,2Nd\wtLd/|ΛN|→1as N→+∞.

We introduce the following sequence of random variables

 XN(ω)=2−Nd(Eω(ΛN,nN,SN)+(B+C)nN), (4.13)

where is the constant from (SI) and is the constant from (LB). By (4.11) this sequence satisfies the inequality

 XN+1(ω)≤2−d2d∑i=1XN(τγ(N)i(ω))+GN

with

 GN=Aρ2\wtL2d2(N+2)d−1R−λN.

In order to show the convergence of the sequence , we establish the following proposition.

{proposition}

Let be a sequence of nonnegative random variables on a probability space , , such that for each there exists a family of probability preserving transformations of , , , with , such that the variables , are i.i.d. (independent identically distributed). If the sequence satisfies

 XN+1(ω)≤1\cardIN∑i∈INXN(τ(N)iω)+GN,where+∞∑N=1|GN|<+∞, (4.14)

then there exists a constant (that does not depend on ) such that

 XNL2−−−−−→N→+∞¯¯¯¯¯X. (4.15)
###### Proof.

Since the terms at the r.h.s. of (4.14) are identically distributed, after taking the expectation one obtains:

 \bbEXN+1≤\bbEXN+GN. (4.16)

Consider the sequence

 cN=\bbEXN−N−1∑i=1Gi.

Obviously, (4.16) guarantees that . Consequently, this sequence converges: . Thus, as soon as the sum also converges, admits the limit that we denote by :

 \bbEXN−−−−−→N→+∞¯¯¯¯¯X.

Consider now the variance of ,