# Direct Scaling Analysis of Fermionic Multiparticle Correlated Anderson Models with Infinite-Range Interaction

Authors: Victor Chulaevsky
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID.


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## Body

## 1. Introduction

### 1.1. The Model: A Brief Historical Overview

Analysis of localization phenomena in multiparticle quantum systems with nontrivial interaction in a random environment is a relatively new direction in the Anderson localization theory, where during almost half a century, since the seminal paper by Anderson [1], most efforts were concentrated on the study of disordered systems in the single-particle approximation, that is, without interparticle interaction. While in numerous physical models such an approximation seems fairly reasonable, it was pointed out already in the first works by Anderson that the multiparticle models presented a real challenge.The number of results on multiparticle localization obtained in the physical and mathematical communities still remains rather limited. We do not review here results obtained by physicists (cf. [2, 3]), based on the methods of theoretical physics and considered as firmly established by the physical community. The rigorous mathematical results on multiparticle localization obtained so far (cf., e.g., [4–7]) apply to N-particle systems with arbitrary, but fixed N>1, and the range of parameters (such as the amplitude of the disorder and/or the proximity to the edge(s) of the spectrum) is rapidly degrading as N→∞.In this paper, we study spectral properties of the multiparticle random Hamiltonians of the form(1)Hω=H0+gVω+U,g∈R,describing N>1 fermionic particles in the configuration space Z assumed to be a finite or countable graph. For clarity, we present first our method in the particular case where Z=Z1 and then describe how to adapt the arguments to more general graphs Z with polynomial growth of balls (cf. Section 8). In (1), H0 is a finite-difference operator representing the kinetic energy, for example, the nearest-neighbor discrete negative Laplacian (-Δ), U is the operator of multiplication by the interaction potential x↦U(x), and V(ω) is the operator of multiplication by the random function (2)x⟼Vx;ω=Vx1;ω+⋯+VxN;ω, where V:Z×Ω→R is a random field relative to some probability space (Ω,F,P). g>0 is a parameter measuring the amplitude of disorder. (All notations are explained in detail in Section 2.)We consider only the restriction ofH(ω) to the fermionic subspace. The bosonic subspace can be treated essentially in the same way.


### 1.2. Motivation for This Work

The two most popular methods of the rigorous Anderson localization theory, the Multiscale Analysis developed in [8–11] and the Fractional Moments Method developed in [12, 13], start with the analysis of the Green functions (GFs) of a given random operator, and then one has to translate the upper bounds on the GFs obtained for a fixed energy (cf. [8, 11–13]) or in an energy interval (cf. [8, 10]) into the language of the eigenfunctions (EFs) and eigenfunction correlators (EFCs). In a prior work [14] we proposed an alternative approach, called the Direct Scaling Analysis of eigenfunctions, which focuses essentially on theeigenfunctions in finite domains of the configuration space, from the very beginning. It keeps some important advantages of the MSA, namely, a greater tolerance than in FMM to lower regularity of the disorder, and is close in spirit to the KAM (Kolmogorov-Arnold-Moser) approach, which often requires a tedious inductive scheme. In fact, work [14] has been inspired by the KAM approach to quasi-periodic Hamiltonians proposed earlier by Sinai [15]. Compared to the KAM method, the DSA has a simpler structure of the scale induction, similar to that of the MSA, although it also inherits somewhat weaker probabilistic bounds typical of the MSA.Furthermore, as was said, there are two kinds of the MSA: with a fixed energy (FEMSA) and with a variable energy (VEMSA); the latter is often referenced to as the energy-interval MSA, for it treats the decay properties of the Green functions in an entire interval of energies at once. The FEMSA is slightly simpler than VEMSA in the single-particle case, but in the realm of multiparticle random Hamiltonians with a nontrivial interaction, the variable-energy variant of the (multiparticle) MSA is considerably more complex. Some of the components of the VEMPMSA are simply eliminated with the (MP)DSA approach, and the counterparts of some other components become quite elementary. A good example of such kind is Lemma21 (see Section 4.3), the proof of which is very simple and fits in a few lines, while its analog in the variable-energy MPMSA (cf., e.g., [4, Lemma 3]) is substantially more involved.This constatation has been the principal motivation for the present work. A reader familiar with the techniques of the papers [4, 16, 17] can see that the multiparticle variant of the DSA (MPDSA) from the present paper is much closer in its logical structure to the FEMSA, yet it leads directly to the VEMSA-type decay bounds on the entire eigenbases in arbitrarily large finite domains.In preprint [18], which has been a starting point for the present paper, we gave the first proof of exponential decay of multiparticle eigenfunctions in presence of an infinite-range interaction decaying at infinity at a fractional-exponential rate, r↦e-rζ, 0<ζ<1. In a recent work [7], Aizenman and Warzel, further developing the techniques of [6], proved fractional-exponential decay of EF correlators (EFCs) for subexponentially decaying interactions of infinite range. Due to the logical structure of the FMM, the decay analysis of the eigenfunctions is subordinate to that of the EFCs, and this explains why [7] established only a subexponential decay of eigenfunctions. The DSA is free from this limitation, and this allows us to prove a genuine exponential decay of the multiparticle EFs even in the case where the interaction decay is subexponential.


### 1.3. Novelty

Apart from simplifications of the Multiparticle Multiscale Analysis (MPMSA) developed in [4], the novelty of this paper is that the external random potential is not necessarily IID (with independent and identically distributed values) but is strongly mixing. Earlier publications, including [19], required the common external random potential acting on the particles to be IID.


### 1.4. Structure of the Paper

The structure of the paper is as follows:(i)
The principal assumptions and main results are stated in Sections2.4–2.6.(ii)
The deterministic techniques of scaling analysis are presented in Section3.(iii)
In Section4, we treat first the case of finite-range interactions, in order to present the logical structure and the methodological advantages of the MPDSA in the most transparent way.(iv)
In Section7, our analysis is adapted to the interactions of infinite range.(v)
Sections5 and 6 are devoted to the derivation of spectral and dynamical localization from the final results of the MPDSA.


## 1.1. The Model: A Brief Historical Overview

Analysis of localization phenomena in multiparticle quantum systems with nontrivial interaction in a random environment is a relatively new direction in the Anderson localization theory, where during almost half a century, since the seminal paper by Anderson [1], most efforts were concentrated on the study of disordered systems in the single-particle approximation, that is, without interparticle interaction. While in numerous physical models such an approximation seems fairly reasonable, it was pointed out already in the first works by Anderson that the multiparticle models presented a real challenge.The number of results on multiparticle localization obtained in the physical and mathematical communities still remains rather limited. We do not review here results obtained by physicists (cf. [2, 3]), based on the methods of theoretical physics and considered as firmly established by the physical community. The rigorous mathematical results on multiparticle localization obtained so far (cf., e.g., [4–7]) apply to N-particle systems with arbitrary, but fixed N>1, and the range of parameters (such as the amplitude of the disorder and/or the proximity to the edge(s) of the spectrum) is rapidly degrading as N→∞.In this paper, we study spectral properties of the multiparticle random Hamiltonians of the form(1)Hω=H0+gVω+U,g∈R,describing N>1 fermionic particles in the configuration space Z assumed to be a finite or countable graph. For clarity, we present first our method in the particular case where Z=Z1 and then describe how to adapt the arguments to more general graphs Z with polynomial growth of balls (cf. Section 8). In (1), H0 is a finite-difference operator representing the kinetic energy, for example, the nearest-neighbor discrete negative Laplacian (-Δ), U is the operator of multiplication by the interaction potential x↦U(x), and V(ω) is the operator of multiplication by the random function (2)x⟼Vx;ω=Vx1;ω+⋯+VxN;ω, where V:Z×Ω→R is a random field relative to some probability space (Ω,F,P). g>0 is a parameter measuring the amplitude of disorder. (All notations are explained in detail in Section 2.)We consider only the restriction ofH(ω) to the fermionic subspace. The bosonic subspace can be treated essentially in the same way.


## 1.2. Motivation for This Work

The two most popular methods of the rigorous Anderson localization theory, the Multiscale Analysis developed in [8–11] and the Fractional Moments Method developed in [12, 13], start with the analysis of the Green functions (GFs) of a given random operator, and then one has to translate the upper bounds on the GFs obtained for a fixed energy (cf. [8, 11–13]) or in an energy interval (cf. [8, 10]) into the language of the eigenfunctions (EFs) and eigenfunction correlators (EFCs). In a prior work [14] we proposed an alternative approach, called the Direct Scaling Analysis of eigenfunctions, which focuses essentially on theeigenfunctions in finite domains of the configuration space, from the very beginning. It keeps some important advantages of the MSA, namely, a greater tolerance than in FMM to lower regularity of the disorder, and is close in spirit to the KAM (Kolmogorov-Arnold-Moser) approach, which often requires a tedious inductive scheme. In fact, work [14] has been inspired by the KAM approach to quasi-periodic Hamiltonians proposed earlier by Sinai [15]. Compared to the KAM method, the DSA has a simpler structure of the scale induction, similar to that of the MSA, although it also inherits somewhat weaker probabilistic bounds typical of the MSA.Furthermore, as was said, there are two kinds of the MSA: with a fixed energy (FEMSA) and with a variable energy (VEMSA); the latter is often referenced to as the energy-interval MSA, for it treats the decay properties of the Green functions in an entire interval of energies at once. The FEMSA is slightly simpler than VEMSA in the single-particle case, but in the realm of multiparticle random Hamiltonians with a nontrivial interaction, the variable-energy variant of the (multiparticle) MSA is considerably more complex. Some of the components of the VEMPMSA are simply eliminated with the (MP)DSA approach, and the counterparts of some other components become quite elementary. A good example of such kind is Lemma21 (see Section 4.3), the proof of which is very simple and fits in a few lines, while its analog in the variable-energy MPMSA (cf., e.g., [4, Lemma 3]) is substantially more involved.This constatation has been the principal motivation for the present work. A reader familiar with the techniques of the papers [4, 16, 17] can see that the multiparticle variant of the DSA (MPDSA) from the present paper is much closer in its logical structure to the FEMSA, yet it leads directly to the VEMSA-type decay bounds on the entire eigenbases in arbitrarily large finite domains.In preprint [18], which has been a starting point for the present paper, we gave the first proof of exponential decay of multiparticle eigenfunctions in presence of an infinite-range interaction decaying at infinity at a fractional-exponential rate, r↦e-rζ, 0<ζ<1. In a recent work [7], Aizenman and Warzel, further developing the techniques of [6], proved fractional-exponential decay of EF correlators (EFCs) for subexponentially decaying interactions of infinite range. Due to the logical structure of the FMM, the decay analysis of the eigenfunctions is subordinate to that of the EFCs, and this explains why [7] established only a subexponential decay of eigenfunctions. The DSA is free from this limitation, and this allows us to prove a genuine exponential decay of the multiparticle EFs even in the case where the interaction decay is subexponential.


## 1.3. Novelty

Apart from simplifications of the Multiparticle Multiscale Analysis (MPMSA) developed in [4], the novelty of this paper is that the external random potential is not necessarily IID (with independent and identically distributed values) but is strongly mixing. Earlier publications, including [19], required the common external random potential acting on the particles to be IID.


## 1.4. Structure of the Paper

The structure of the paper is as follows:(i)
The principal assumptions and main results are stated in Sections2.4–2.6.(ii)
The deterministic techniques of scaling analysis are presented in Section3.(iii)
In Section4, we treat first the case of finite-range interactions, in order to present the logical structure and the methodological advantages of the MPDSA in the most transparent way.(iv)
In Section7, our analysis is adapted to the interactions of infinite range.(v)
Sections5 and 6 are devoted to the derivation of spectral and dynamical localization from the final results of the MPDSA.


## 2. Preliminaries, Assumptions, and Main Results

### 2.1. Configurations of Indistinguishable Particles inZ1

In the first part of this paper, we work with configurations ofN≥1 quantum particles in the one-dimensional lattice, Z=Z1. In quantum mechanics, particles of the same kind are considered indistinguishable; more precisely, depending on the nature of the particles, the wave functions describing N>1 particles have to be either symmetric (Bose-Einstein quantum statistics) or antisymmetric (Fermi-Dirac quantum statistics). We choose here the fermionic case; this gives rise to slightly simpler notations and constructions.For clarity, we use often boldface symbols for the objects related to the multiparticle systems.Quantum states of anN-particle fermionic system in Z are elements of the Hilbert space HN=HN,-, formed by all square-summable antisymmetric functions Ψ:ZN→C, with the inner product 〈·∣·〉 inherited from the Hilbert space l2(ZN)=H1⊗N. In this particular case where the “physical” configuration space is one-dimensional, HN admits a simple representation which we will use.First, note that any antisymmetric function(3)x=x1,…,xN⟼Ψx1,…,xNvanishes on all hyperplanes {x:xi=xj}, 1≤i<j≤N. Further, any function defined on the “positive sector” Z>N={(x1,…,xN):x1>x2>⋯>xN} admits a unique antisymmetric continuation to ZN≔ZN. An orthonormal basis in HN can be chosen in the usual form (4)Φax=1N!∑π∈SN-1π⨂j=1N1aπ-1j,where a∈ZN with #{a1,…,aN}=N and 1a is the indicator function of the single-point set {a}. Here SN is the symmetric group acting in ZN by permutations of the particle positions, π(a)=(aπ-1(1),…,aπ-1(N)), and (-1)π∈{+1,-1} denotes the parity of the permutation π. It is readily seen that HN is unitarily isomorphic to the Hilbert space l2(Z>N) of square-summable functions on Z>N; equivalently, one can consider square-summable functions on the set Z≥N={x1≥⋯≥xN} vanishing on the boundary Z=N={x∈Z≥N:∃i≠j,xi=xj}. Indeed, the isomorphism is induced by the bijection between the orthonormal bases {Φa} and {1a}: (5)Φa⟷1a=1a1⊗⋯⊗1aN,a∈Z>N.The subspace HN is invariant under any operator commuting with the action of the symmetric group SN.An advantage of the above representation is thatZ>N inherits its explicit, natural graph structure from ZN. For more general graphs Z, one has to resort to the general construction of the so-called symmetric powers of graphs; see the details in Section 8.


### 2.2. Fermionic Laplacians

Any unordered finite or countable connected graph(G,E) with the set of vertices G and the set of edges E is endowed with the canonical graph distance (x,y)↦dG(x,y) (defined as the length f the shortest path x⇝y over the edges) and with the canonical graph Laplacian ΔZ: (6)ΔGfx=∑x,yfy-fx=-nGxfx+∑x,yfy;here 〈x,y〉 denotes a pair of nearest neighbors, and nG(x) is the coordination number of the point x in G, that is, the number of its nearest neighbors.In particular, one can takeG=ZN (or a subgraph thereof) with the edges 〈x,y〉 formed by vertices x, y with |x-y|1≔|x1-y1|+⋯+|xN-yN|=1. In other words, the vector norm |·|1 induces the graph distance on ZN. The Laplacian on ZN, which we will now denote by Δ, can be written as follows: (7)Δ=∑j=1N⨂i=1j-11i⊗Δj⊗⨂k=j+1N1k,where 1(i) is the identity operator acting on the ith variable and Δ(j) is the one-dimensional negative lattice Laplacian in the jth variable: (8)Δjfxj=-2fxj+fxj-1+fxj+1,xj∈Z.The restriction to the subspace HN of antisymmetric functions can be equivalently defined in terms of functions supported by the positive sector Z>N, hence vanishing on its border Z=N. Indeed, the matrix elements of Δ in the basis Φa can be nonzero only for pairs Φa, Φb with |a-b|1=1, so that, for some j∈[1,N], |aj-bj|=1, while for all i≠j, ai=bi. If a,b∈Z>N, then 〈Φa|Δ|Φb〉=1/N!〈1a|Δ|1b〉. If, say, a∈Z=N, then the respective matrix element of the Laplacian’s restriction to Z>N with Dirichlet boundary conditions on Z=N vanishes, and so does the function Φa (which is no longer an element of the basis in HN). Therefore, up to a constant factor, the restriction of the N-particle Laplacian to the fermionic subspace HN is unitarily equivalent to the standard graph Laplacian on Z>N. From this point on, we will work with the latter.It will be convenient in the course of the scaling analysis to use a different kind of metric onZ>N⊂ZN, the max-distance defined by (9)ρx,y=max1≤j≤NdZxj,yj,and to work with the balls relative to the max-distance,(10)BLx=y:ρx,y≤L=⨉j=1NBLxj.Here BL(x)=[x-L,x+L]∩Z.In the positive sectorZ>N, the above factorization of ρ-balls is subject to the condition that the RHS of (10) is itself a subset of the positive sector Z>N (but the inclusion “⊂” always holds true). Considering configurations x={x1,…,xn} as subsets of Z, one can define the distance between two configurations x′∈Z>(n′) and x′′∈Z>(n′′) in a usual way: (11)distx′,x′′=minu∈x′minv∈x′′u-v.Lemma 1.
Letx∈Z>N be a union of two subconfigurations x′∈Z>(n′) and x′′∈Z>(n′′), such that dist(x′,x′′)>2L. Then the following identity holds true: (12)BLNx=BLn′x′×BLn′′x′′.The proof is straightforward and will be omitted.The graph distancedZ>N will be useful in some geometrical constructions and definitions, referring to the graph structure inherited from ZN.Given a subgraphΛ⊂Z>N, we define its internal, external, and the so-called graph (or edge) boundary, in terms of the canonical graph distance (below Λc stands for the complement of Λ): (13)∂-Λ=y∈Λ:dZ>Ny,Λc=1,∂+Λ=∂-Λc∂Λ=x,y∈Λ×Λc:dZ>Nx,y=1.We also define the occupation numbers of a sitey in the 1-particle configuration space relative to a configuration x={x1,…,xN}. Namely, define a function nx:Z↦N by nx(y)=#{j:xj=y}, y∈X.


### 2.3. Multiparticle Fermionic Hamiltonians

Given a finite subsetΛ⊂Z>N, let Σ(HΛ) be the spectrum of the operator HΛ and GΛ(E)=(HΛ-E)-1 its resolvent. The matrix elements of resolvents for E∉Σ(HΛ), in the canonical delta-basis, usually referred to as the Green functions, are denoted by GΛ(x,y;E). In the context of random operators, the dependence upon the element ω∈Ω will be often omitted for brevity, unless required or instructive. Similar notations will be used for infinite Λ.Given a real-valued functionW:Z>N→R, we identify it with the operator of multiplication by W and consider the fermionic N-particle random Hamiltonian (14)Hω=H0+W, where H0 is a second-order finite-difference operator, for example, the negative graph Laplacian (-Δ). Furthermore, in our model the potential energy is random: (15)Wx=Wx;ω=gVx;ω+Ux,where gV(·;ω) is the external random potential energy of the form (16)gVx;ω=gVx1;ω+⋯+gVxN;ω,of amplitude g>0, and V:Z×Ω→R is a random field on Z relative to a probability space (Ω,F,P). The expectation relative to the measure P will be denoted by E·. Our assumptions on V are listed below (cf. (W1)–(W3)). See also Remark 4 where how (W1)–(W3) can be relaxed is explained.Further,U is the interaction energy operator; for notational brevity, we assume that it is generated by a two-body interaction potential U, (17)Ux=∑i≠jUxi-xj, satisfying one of hypotheses (U0)-(U1) (see below).Note that, without loss of generality, ifH0<∞ and V is a.s. bounded, then the random operator has a.s. bounded norm; hence its spectrum is a.s. covered by some bounded interval I⊂R. In Remark 3, we explain that the boundedness of the potential is not crucial for the spectral localization, although the statement of the result on the dynamical localization is to be slightly modified for unbounded potentials.One can also consider higher-order finite-difference operatorsH0; this requires only minor technical modifications.


### 2.4. Assumptions on the Random Potential

We formulate the assumptions in the general situation where the1-particle configuration space is a graph Z with graph-distance dZ; this includes the case Z=Z1.We assume that the random fieldV:Z×Ω→R is (possibly) correlated, but strongly mixing; this includes of course the IID potentials treated earlier in [4, 6]. Let (18)FV,xt=PVx;ω≤t,x∈Z be the marginal probability distribution functions (PDF) of the random field V and (19)FV,xt∣F≠x≔PVx;ω≤t∣F≠xthe conditional distribution functions (CDF) of the random field V given the sigma-algebra F≠xV generated by random variables {V(y;ω),y≠x}. Our assumptions on correlated potentials are as follows:(W1)
The marginal CDFs are uniformly Lipschitz continuous: for someC∈(0,+∞) and all s∈(0,1](20)supx∈Zsupa∈RFV,xa+s∣F≠xV-FV,xa∣F≠xV≤Cs.
The LHS of (20) is a conditional probability, hence, a random variable, so (20) holds P-a.s.To formulate the next assumption, introduce the following notation: given a subsetΛ⊂Z, we denote by FΛV the sigma-algebra generated by the values of the random potential {V(x;ω),x∈Λ}.(W2)
(Rosenblatt strong mixing) For any pair of subsets B′, B′′⊂Z with dZ(B′,B′′)≥L≥1, any events E′∈FB′V, E′′∈FB′′V and some C>0(21)PE′∩E′′-PE′PE′′≤e-Cln2L.One can easily check that, for anyp>0, α∈(1,2), some ϑ∈(0,1), a>0, and L0∈N large enough,(22)e-Cln2L0αk<L0αk-ap1+θk,k≥0.The rate of decay of correlations indicated in the RHS of (22) is required to prove dynamical localization bounds with decay rate of EF correlators faster than polynomial, and it can be relaxed to a power-law decay, if one aims to prove only a power-law decay of EF correlators.Assumptions (W1)-(W2) are sufficient for the proof of spectral and strong dynamical localization in multiparticle systems. In particular, the role of (W1) is to guarantee the eigenvalue concentration (EVC) estimates used in the MPMSA scheme. However, it was discovered in [4, 6] that the traditional EVC estimates do not provide all necessary information forefficient decay bounds on the eigenfunctions of multiparticle operators. More precisely, conventional EVC bounds seem so far insufficient for the proof of the exponential decay of eigenfunctions and EF correlators with respect to a norm in the configuration space of N-particle systems, starting from N=3. For this reason, we proposed earlier [20] a new method for comparing spectra of two strongly correlated multiparticle subsystems and improved the EVC estimate from [4]. For the new method to apply, one needs an additional assumption on the random potential field which we will describe now.First, introduce the following notations. Given a finite subsetQ⊂Z, we denote by ξQ(ω) the sample mean of the random field V over Q, (23)ξQω≔Q-1∑x∈QVx,ω, and define the “fluctuations” of V relative to the sample mean, ηx≔V(x,ω)-ξQ(ω), x∈Q. Denote by FQ the sigma-algebra generated by {ηx,Vy:x∈Q,y∉Q}.We will assume that the random fieldV fulfills the following condition:(W3)
There existC′,A′,b′∈(0,+∞) such that for any finite subset Q⊂Z of cardinality |Q| and any FQ-measurable function λ and all s∈(0,1](24)PξQω∈λω,λω+s≤C′QA′sb′.In the particular case of a Gaussian IID field, for example, with zero mean and unit variance, it is well-known from the standard courses of probability thatξQ is independent of the “fluctuations” ηx, so it is Gaussian with variance |Q|-1 (if V(x;ω) has unit variance) that its probability density pQ is bounded, although pQ∞~|Q|1/2→∞ as |Q|→∞. Property (W3) has been recently proven for a larger class of IID potentials (cf. [21]).Specifically, (W3) has been established in [21] for IID random potentials with marginal probability measure μ which admits a representation of the form μ=μ1∗μ2, where μ2 is an arbitrary probability measure on R and μ1 is supported by some bounded interval [a,b] and admits probability density ρ1(·) which fulfills the following condition: (25)0<ρ∗≤ρ1t≤ρ¯<∞,ρ1′tρ1t≤C1<∞,∀t∈a,b.A prototypical example is the uniform distribution on [a,b].


### 2.5. Assumptions on the Interaction Potential

We assume that the interaction potentialU generating the interaction energy U satisfies one of the following decay conditions:(U0)
There existsr0<∞ such that (26)Ur=0,∀r≥r0.(U1)
There are someC∈(0,+∞), δ∈(0,1/14), and ζ∈(0,δ/1+δ) such that (27)Ur≤Ce-cr1-ζ,∀r≥0.Naturally, (U0) ⇒ (U1). We will prove the multiparticle localization first under the strongest assumption (U0) (leading to a simpler proof), to illustrate the general structure of the DSA procedure, and then extend the proof to the infinite-range interactions satisfying (U1).


### 2.6. Main Results

Theorem 2.
Assume that a bounded random fieldV fulfills conditions (W1)–(W3), and the interaction U fulfills one of the conditions (U0) and (U1). There exists g0∈(0,+∞) such that for |g|≥g0 the following holds true. (A)
With probability one, the fermionic operatorH(ω)=-Δ+gV(ω)+U has pure point spectrum and all its eigenfunctions Ψj(ω) are exponentially decaying at infinity: for each Ψj, some x^j, and all x with ρ(x^j,x) large enough,(28)Ψjx,ω≤e-mρx^j,x,m>0.(B)
For someC,a,c>0 and any x,y∈Z>N(29)Esupt∈R1ye-itHω1x≤Ce-aln1+cρx,y.Consequently, for any finite subsetK⊂Z>N(30)Esupt∈Rea/2ln1+cXe-itHω1K<∞,whereXK is the operator of multiplication by the function x↦(ρ(K,x)+1).Remark 3.
The assumption of boundedness of the random potential is inessential for the proof of spectral localization (exponential decay of all eigenfunctions) and for the analog of the decay bound (29) on eigenfunction correlators in arbitrary large but finite subset Λ⊂Z>N of the N-particle configuration space (cf. Corollary 27). Without any condition on the rate of decay of the tail probabilities P|V(x;ω)|≥s as s→+∞, the propagator e-itH(ω) in (29)-(30) has to be replaced with PI(H(ω))e-itH(ω), where I⊂R is an arbitrary bounded interval, and PI(H(ω)) is the spectral projection for H(ω) on I.Remark 4.
Exponential spectral localization can be proved without condition (W3), with (W1) relaxed to Hölder continuity. Indeed, one can use the EVC bound from [4, Lemma 2], valid for the pairs of balls distant in the so-called Hausdorff pseudo-metric dH in the multiparticle configuration space, defined as follows: for the configurations x={x1,…,xN}, y={y1,…,yN}, (31)dHx,y≔maxmaxidZxi,y,maxidZyi,x, where (32)dZx,y=minjdZx,yj.In this case, the decay bounds on the eigenfunction correlators would also be established with respect to dH replacing the max-distance ρ:(33)Esupt∈R1ye-itHω1x≤Cxe-aln1+cdHx,y.Observe that the RHS bound is not uniform as in (29) but depends upon x; this is an artefact of the EVC bounds based on the Hausdorff distance (cf. [4, 6, 7, 17]). Of course, the roles of x and y in the LHS of (33) are similar, so the factor C(x) in the RHS can be replaced by C(y) or, more precisely, by some function C′(diam(x)∧diam(y)). As to the proofs, they require only minor notational modifications, for they adapt easily to any kind of pseudometric in the N-particle configuration space for which an EVC bound of the general form (64) can be proved, with some hL(s)≤CLAsn, C,A∈(0,+∞), and b∈(0,1].


## 2.1. Configurations of Indistinguishable Particles inZ1

In the first part of this paper, we work with configurations ofN≥1 quantum particles in the one-dimensional lattice, Z=Z1. In quantum mechanics, particles of the same kind are considered indistinguishable; more precisely, depending on the nature of the particles, the wave functions describing N>1 particles have to be either symmetric (Bose-Einstein quantum statistics) or antisymmetric (Fermi-Dirac quantum statistics). We choose here the fermionic case; this gives rise to slightly simpler notations and constructions.For clarity, we use often boldface symbols for the objects related to the multiparticle systems.Quantum states of anN-particle fermionic system in Z are elements of the Hilbert space HN=HN,-, formed by all square-summable antisymmetric functions Ψ:ZN→C, with the inner product 〈·∣·〉 inherited from the Hilbert space l2(ZN)=H1⊗N. In this particular case where the “physical” configuration space is one-dimensional, HN admits a simple representation which we will use.First, note that any antisymmetric function(3)x=x1,…,xN⟼Ψx1,…,xNvanishes on all hyperplanes {x:xi=xj}, 1≤i<j≤N. Further, any function defined on the “positive sector” Z>N={(x1,…,xN):x1>x2>⋯>xN} admits a unique antisymmetric continuation to ZN≔ZN. An orthonormal basis in HN can be chosen in the usual form (4)Φax=1N!∑π∈SN-1π⨂j=1N1aπ-1j,where a∈ZN with #{a1,…,aN}=N and 1a is the indicator function of the single-point set {a}. Here SN is the symmetric group acting in ZN by permutations of the particle positions, π(a)=(aπ-1(1),…,aπ-1(N)), and (-1)π∈{+1,-1} denotes the parity of the permutation π. It is readily seen that HN is unitarily isomorphic to the Hilbert space l2(Z>N) of square-summable functions on Z>N; equivalently, one can consider square-summable functions on the set Z≥N={x1≥⋯≥xN} vanishing on the boundary Z=N={x∈Z≥N:∃i≠j,xi=xj}. Indeed, the isomorphism is induced by the bijection between the orthonormal bases {Φa} and {1a}: (5)Φa⟷1a=1a1⊗⋯⊗1aN,a∈Z>N.The subspace HN is invariant under any operator commuting with the action of the symmetric group SN.An advantage of the above representation is thatZ>N inherits its explicit, natural graph structure from ZN. For more general graphs Z, one has to resort to the general construction of the so-called symmetric powers of graphs; see the details in Section 8.


## 2.2. Fermionic Laplacians

Any unordered finite or countable connected graph(G,E) with the set of vertices G and the set of edges E is endowed with the canonical graph distance (x,y)↦dG(x,y) (defined as the length f the shortest path x⇝y over the edges) and with the canonical graph Laplacian ΔZ: (6)ΔGfx=∑x,yfy-fx=-nGxfx+∑x,yfy;here 〈x,y〉 denotes a pair of nearest neighbors, and nG(x) is the coordination number of the point x in G, that is, the number of its nearest neighbors.In particular, one can takeG=ZN (or a subgraph thereof) with the edges 〈x,y〉 formed by vertices x, y with |x-y|1≔|x1-y1|+⋯+|xN-yN|=1. In other words, the vector norm |·|1 induces the graph distance on ZN. The Laplacian on ZN, which we will now denote by Δ, can be written as follows: (7)Δ=∑j=1N⨂i=1j-11i⊗Δj⊗⨂k=j+1N1k,where 1(i) is the identity operator acting on the ith variable and Δ(j) is the one-dimensional negative lattice Laplacian in the jth variable: (8)Δjfxj=-2fxj+fxj-1+fxj+1,xj∈Z.The restriction to the subspace HN of antisymmetric functions can be equivalently defined in terms of functions supported by the positive sector Z>N, hence vanishing on its border Z=N. Indeed, the matrix elements of Δ in the basis Φa can be nonzero only for pairs Φa, Φb with |a-b|1=1, so that, for some j∈[1,N], |aj-bj|=1, while for all i≠j, ai=bi. If a,b∈Z>N, then 〈Φa|Δ|Φb〉=1/N!〈1a|Δ|1b〉. If, say, a∈Z=N, then the respective matrix element of the Laplacian’s restriction to Z>N with Dirichlet boundary conditions on Z=N vanishes, and so does the function Φa (which is no longer an element of the basis in HN). Therefore, up to a constant factor, the restriction of the N-particle Laplacian to the fermionic subspace HN is unitarily equivalent to the standard graph Laplacian on Z>N. From this point on, we will work with the latter.It will be convenient in the course of the scaling analysis to use a different kind of metric onZ>N⊂ZN, the max-distance defined by (9)ρx,y=max1≤j≤NdZxj,yj,and to work with the balls relative to the max-distance,(10)BLx=y:ρx,y≤L=⨉j=1NBLxj.Here BL(x)=[x-L,x+L]∩Z.In the positive sectorZ>N, the above factorization of ρ-balls is subject to the condition that the RHS of (10) is itself a subset of the positive sector Z>N (but the inclusion “⊂” always holds true). Considering configurations x={x1,…,xn} as subsets of Z, one can define the distance between two configurations x′∈Z>(n′) and x′′∈Z>(n′′) in a usual way: (11)distx′,x′′=minu∈x′minv∈x′′u-v.Lemma 1.
Letx∈Z>N be a union of two subconfigurations x′∈Z>(n′) and x′′∈Z>(n′′), such that dist(x′,x′′)>2L. Then the following identity holds true: (12)BLNx=BLn′x′×BLn′′x′′.The proof is straightforward and will be omitted.The graph distancedZ>N will be useful in some geometrical constructions and definitions, referring to the graph structure inherited from ZN.Given a subgraphΛ⊂Z>N, we define its internal, external, and the so-called graph (or edge) boundary, in terms of the canonical graph distance (below Λc stands for the complement of Λ): (13)∂-Λ=y∈Λ:dZ>Ny,Λc=1,∂+Λ=∂-Λc∂Λ=x,y∈Λ×Λc:dZ>Nx,y=1.We also define the occupation numbers of a sitey in the 1-particle configuration space relative to a configuration x={x1,…,xN}. Namely, define a function nx:Z↦N by nx(y)=#{j:xj=y}, y∈X.


## 2.3. Multiparticle Fermionic Hamiltonians

Given a finite subsetΛ⊂Z>N, let Σ(HΛ) be the spectrum of the operator HΛ and GΛ(E)=(HΛ-E)-1 its resolvent. The matrix elements of resolvents for E∉Σ(HΛ), in the canonical delta-basis, usually referred to as the Green functions, are denoted by GΛ(x,y;E). In the context of random operators, the dependence upon the element ω∈Ω will be often omitted for brevity, unless required or instructive. Similar notations will be used for infinite Λ.Given a real-valued functionW:Z>N→R, we identify it with the operator of multiplication by W and consider the fermionic N-particle random Hamiltonian (14)Hω=H0+W, where H0 is a second-order finite-difference operator, for example, the negative graph Laplacian (-Δ). Furthermore, in our model the potential energy is random: (15)Wx=Wx;ω=gVx;ω+Ux,where gV(·;ω) is the external random potential energy of the form (16)gVx;ω=gVx1;ω+⋯+gVxN;ω,of amplitude g>0, and V:Z×Ω→R is a random field on Z relative to a probability space (Ω,F,P). The expectation relative to the measure P will be denoted by E·. Our assumptions on V are listed below (cf. (W1)–(W3)). See also Remark 4 where how (W1)–(W3) can be relaxed is explained.Further,U is the interaction energy operator; for notational brevity, we assume that it is generated by a two-body interaction potential U, (17)Ux=∑i≠jUxi-xj, satisfying one of hypotheses (U0)-(U1) (see below).Note that, without loss of generality, ifH0<∞ and V is a.s. bounded, then the random operator has a.s. bounded norm; hence its spectrum is a.s. covered by some bounded interval I⊂R. In Remark 3, we explain that the boundedness of the potential is not crucial for the spectral localization, although the statement of the result on the dynamical localization is to be slightly modified for unbounded potentials.One can also consider higher-order finite-difference operatorsH0; this requires only minor technical modifications.


## 2.4. Assumptions on the Random Potential

We formulate the assumptions in the general situation where the1-particle configuration space is a graph Z with graph-distance dZ; this includes the case Z=Z1.We assume that the random fieldV:Z×Ω→R is (possibly) correlated, but strongly mixing; this includes of course the IID potentials treated earlier in [4, 6]. Let (18)FV,xt=PVx;ω≤t,x∈Z be the marginal probability distribution functions (PDF) of the random field V and (19)FV,xt∣F≠x≔PVx;ω≤t∣F≠xthe conditional distribution functions (CDF) of the random field V given the sigma-algebra F≠xV generated by random variables {V(y;ω),y≠x}. Our assumptions on correlated potentials are as follows:(W1)
The marginal CDFs are uniformly Lipschitz continuous: for someC∈(0,+∞) and all s∈(0,1](20)supx∈Zsupa∈RFV,xa+s∣F≠xV-FV,xa∣F≠xV≤Cs.
The LHS of (20) is a conditional probability, hence, a random variable, so (20) holds P-a.s.To formulate the next assumption, introduce the following notation: given a subsetΛ⊂Z, we denote by FΛV the sigma-algebra generated by the values of the random potential {V(x;ω),x∈Λ}.(W2)
(Rosenblatt strong mixing) For any pair of subsets B′, B′′⊂Z with dZ(B′,B′′)≥L≥1, any events E′∈FB′V, E′′∈FB′′V and some C>0(21)PE′∩E′′-PE′PE′′≤e-Cln2L.One can easily check that, for anyp>0, α∈(1,2), some ϑ∈(0,1), a>0, and L0∈N large enough,(22)e-Cln2L0αk<L0αk-ap1+θk,k≥0.The rate of decay of correlations indicated in the RHS of (22) is required to prove dynamical localization bounds with decay rate of EF correlators faster than polynomial, and it can be relaxed to a power-law decay, if one aims to prove only a power-law decay of EF correlators.Assumptions (W1)-(W2) are sufficient for the proof of spectral and strong dynamical localization in multiparticle systems. In particular, the role of (W1) is to guarantee the eigenvalue concentration (EVC) estimates used in the MPMSA scheme. However, it was discovered in [4, 6] that the traditional EVC estimates do not provide all necessary information forefficient decay bounds on the eigenfunctions of multiparticle operators. More precisely, conventional EVC bounds seem so far insufficient for the proof of the exponential decay of eigenfunctions and EF correlators with respect to a norm in the configuration space of N-particle systems, starting from N=3. For this reason, we proposed earlier [20] a new method for comparing spectra of two strongly correlated multiparticle subsystems and improved the EVC estimate from [4]. For the new method to apply, one needs an additional assumption on the random potential field which we will describe now.First, introduce the following notations. Given a finite subsetQ⊂Z, we denote by ξQ(ω) the sample mean of the random field V over Q, (23)ξQω≔Q-1∑x∈QVx,ω, and define the “fluctuations” of V relative to the sample mean, ηx≔V(x,ω)-ξQ(ω), x∈Q. Denote by FQ the sigma-algebra generated by {ηx,Vy:x∈Q,y∉Q}.We will assume that the random fieldV fulfills the following condition:(W3)
There existC′,A′,b′∈(0,+∞) such that for any finite subset Q⊂Z of cardinality |Q| and any FQ-measurable function λ and all s∈(0,1](24)PξQω∈λω,λω+s≤C′QA′sb′.In the particular case of a Gaussian IID field, for example, with zero mean and unit variance, it is well-known from the standard courses of probability thatξQ is independent of the “fluctuations” ηx, so it is Gaussian with variance |Q|-1 (if V(x;ω) has unit variance) that its probability density pQ is bounded, although pQ∞~|Q|1/2→∞ as |Q|→∞. Property (W3) has been recently proven for a larger class of IID potentials (cf. [21]).Specifically, (W3) has been established in [21] for IID random potentials with marginal probability measure μ which admits a representation of the form μ=μ1∗μ2, where μ2 is an arbitrary probability measure on R and μ1 is supported by some bounded interval [a,b] and admits probability density ρ1(·) which fulfills the following condition: (25)0<ρ∗≤ρ1t≤ρ¯<∞,ρ1′tρ1t≤C1<∞,∀t∈a,b.A prototypical example is the uniform distribution on [a,b].


## 2.5. Assumptions on the Interaction Potential

We assume that the interaction potentialU generating the interaction energy U satisfies one of the following decay conditions:(U0)
There existsr0<∞ such that (26)Ur=0,∀r≥r0.(U1)
There are someC∈(0,+∞), δ∈(0,1/14), and ζ∈(0,δ/1+δ) such that (27)Ur≤Ce-cr1-ζ,∀r≥0.Naturally, (U0) ⇒ (U1). We will prove the multiparticle localization first under the strongest assumption (U0) (leading to a simpler proof), to illustrate the general structure of the DSA procedure, and then extend the proof to the infinite-range interactions satisfying (U1).


## 2.6. Main Results

Theorem 2.
Assume that a bounded random fieldV fulfills conditions (W1)–(W3), and the interaction U fulfills one of the conditions (U0) and (U1). There exists g0∈(0,+∞) such that for |g|≥g0 the following holds true. (A)
With probability one, the fermionic operatorH(ω)=-Δ+gV(ω)+U has pure point spectrum and all its eigenfunctions Ψj(ω) are exponentially decaying at infinity: for each Ψj, some x^j, and all x with ρ(x^j,x) large enough,(28)Ψjx,ω≤e-mρx^j,x,m>0.(B)
For someC,a,c>0 and any x,y∈Z>N(29)Esupt∈R1ye-itHω1x≤Ce-aln1+cρx,y.Consequently, for any finite subsetK⊂Z>N(30)Esupt∈Rea/2ln1+cXe-itHω1K<∞,whereXK is the operator of multiplication by the function x↦(ρ(K,x)+1).Remark 3.
The assumption of boundedness of the random potential is inessential for the proof of spectral localization (exponential decay of all eigenfunctions) and for the analog of the decay bound (29) on eigenfunction correlators in arbitrary large but finite subset Λ⊂Z>N of the N-particle configuration space (cf. Corollary 27). Without any condition on the rate of decay of the tail probabilities P|V(x;ω)|≥s as s→+∞, the propagator e-itH(ω) in (29)-(30) has to be replaced with PI(H(ω))e-itH(ω), where I⊂R is an arbitrary bounded interval, and PI(H(ω)) is the spectral projection for H(ω) on I.Remark 4.
Exponential spectral localization can be proved without condition (W3), with (W1) relaxed to Hölder continuity. Indeed, one can use the EVC bound from [4, Lemma 2], valid for the pairs of balls distant in the so-called Hausdorff pseudo-metric dH in the multiparticle configuration space, defined as follows: for the configurations x={x1,…,xN}, y={y1,…,yN}, (31)dHx,y≔maxmaxidZxi,y,maxidZyi,x, where (32)dZx,y=minjdZx,yj.In this case, the decay bounds on the eigenfunction correlators would also be established with respect to dH replacing the max-distance ρ:(33)Esupt∈R1ye-itHω1x≤Cxe-aln1+cdHx,y.Observe that the RHS bound is not uniform as in (29) but depends upon x; this is an artefact of the EVC bounds based on the Hausdorff distance (cf. [4, 6, 7, 17]). Of course, the roles of x and y in the LHS of (33) are similar, so the factor C(x) in the RHS can be replaced by C(y) or, more precisely, by some function C′(diam(x)∧diam(y)). As to the proofs, they require only minor notational modifications, for they adapt easily to any kind of pseudometric in the N-particle configuration space for which an EVC bound of the general form (64) can be proved, with some hL(s)≤CLAsn, C,A∈(0,+∞), and b∈(0,1].


## 3. Deterministic Bounds

### 3.1. Geometric Resolvent Inequality

The most essential part of the Direct Scaling Analysis concerns the finite-volume approximationsHΛ(ω) of the random Hamiltonian H(ω), acting in finite-dimensional spaces HΛ, with Λ⊂Z>N, |Λ|≡cardΛ<∞. We define (34)HΛω≔1ΛHω1Λ↾HΛ,where the indicator function 1Λ is identified with the operator of multiplication by 1Λ. HΛ is a bounded, finite-dimensional Hermitian operator, so its spectrum Σ(HΛ) is real.Operator(-Δ) can be represented as follows: (35)-Δ=-nZN+∑x,yΓxy+Γyx,Γxyfx≔δxyfy, and δxy is the Kronecker symbol.Recall that we denote byGΛ(E)=(HΛ-E)-1 the resolvent of HΛ and by G(x,y;E) the matrix elements thereof in the standard delta-basis (the Green functions).The so-called Geometric Resolvent Inequality (GRI) for the Green functions can be easily deduced from the second resolvent identity (we omitE for brevity):(36)GBLx,y≤Clmaxv∈∂-BlxGBlxx,vmaxv′∈∂+BlxGBLv′,y,where(37)Cl=∂Blv≤CNlN.See, for example, [22]. Clearly, (36) implies the inequality(38)GBLx,y≤Clmaxρx,v=lGBlxx,vmaxρu,v′≤l+1GBLv′,y.Sometimes we use another inequality stemming from (36):(39)GBLx,y;E≤ClGBluEmaxv:ρu,v≤l+1GBLv,y;E.Similarly, for the solutions ψ of the eigenfunction equation Hψ=Eψ one has, with x∈Bl(u),(40)ψx≤ClGBlxEmaxy:ρx,y≤l+1ψy,provided that E is not an eigenvalue of the operator HBl(u).


### 3.2. Dominated Decay Bounds for the Green Functions and Eigenfunctions

The analytic statements of this subsection apply indifferently to any discrete Schrödinger operators, on fairly general graphs, regardless of their single- or multiparticle structure of the potential energy. To avoid any confusion, in this subsection we denote the underlying graph byG; in applications to N-particle models in Z, one has to set G=Z>N.Definition 5.
Fix an integerl≥0 and a number q∈(0,1). Consider a finite connected subgraph Λ⊂G and a ball BR(u)⊊Λ. A function f:BR(u)→R+ is called (l,q)-dominated in BR(u) if for any ball Bl(x)⊂ΛR(u) one has(41)fx≤qmaxy:dx,y≤lfy.Lemma 6.
Let be given integersL≥l≥0 and a number q∈(0,1). If f:Λ→R+ on a finite connected graph Λ is (l,q)-dominated in a ball BL(x)⊊Λ, then(42)fx≤qL+1/l+1Mf,B≤qL-l/l+1Mf,B,Mf,Λ≔maxx∈Λfx.Proof.
The claim follows from (41) by induction (“radial descent”): (43)fx≤qMf,Bl+1≤⋯≤qjMf,Bjl+1≤⋯≤qL+1/l+1Mf,B.Lemma 7.
LetΛ be a finite connected graph and f:Λ×Λ→R+, (x′,x′′)↦f(x′,x′′), a function which is separately (l,q)-dominated in x′∈Br′(u′)⊊Λ and in x′′∈Br′′(u′′)⊊Λ, with d(u′,u′′)≥r′+r′′+2. Then(44)fu′,u′′≤qr′+1/l+1+r′′+1/l+1Mf,B≤qr′+r′′-2l/l+1Mf,B.Proof.
Fix any pointy′′∈Br′′+1(u′′) and consider the function (45)fy′′:y′⟼fy′,y′′ which is (l,q)-dominated in Br′(u′)⊊Λ, so by Lemma 6,(46)fy′′u′=fu′,y′′≤qr′+1/l+1Mf,Λ×Λ.Now introduce the function f~u′:y′′↦f(u′,y′′) which is (l,q)-dominated in Br′′(u′′)⊊Λ and bounded by the RHS of (46). Applying again Lemma 6, the claim follows.The relevance of the above notions and results is illustrated by Lemma10, stated in the next subsection. It could be stated in a general form, for arbitrary countable graphs, but its formulation relies on the notions of “nonsingular” and “nonresonant” balls. For our model, these are given in Definitions 8 and 9. However, the proof given in [14] (cf. [14, Lemma 4]) applies indifferently to arbitrary graphs.


### 3.3. Localization and Tunneling in Finite Balls

From this point on, we will work with a sequence of length scales, positive integers{Lk,k≥0} defined recursively by Lk+1=⌈Lkα⌉, L0>2. For clarity, we keep the value α=4/3; observe that α2<2. In several arguments we will require the initial scale L0 to be large enough.Definition 8.
Given a sample of the random potentialV(·;ω), a ball BL(u) is called (i)
E-nonresonant (E-NR) if GB(E;ω)≤e+Lβ and E-resonant (E-R) otherwise;(ii)
completelyE-nonresonant (E-CNR) if it does not contain any E-R ball Bl(u)⊆BL(u) with l≥L1/α and E-partially resonant (E-PR) otherwise.Definition 9.
Given a sampleV(·;ω), a ball BL(u) is called (E,m)-nonsingular ((E,m)-NS) if(47)maxy∈∂-BLuGBLuu,y;E;ω≤e-γm,LL+2Lβ,where(48)γm,L≔m1+L-τ,τ=18.Otherwise, it is called (E,m)-singular ((E,m)-S).Lemma 10 (cf. [14, Lemma 4]).
Consider a ballBLk(u) and an operator H=HBLk(u) with fixed (nonrandom) potential W. Let {ψj,j=1,…,|BLk(u)|} be the normalized eigenfunctions of H. Pick a pair of points x0,y0∈BLk(u) with d(x0,y0)>Lk-1+1 and an integer (49)R∈Lk-1,dx0,y0-Lk-1+1. Suppose that any ball BLk-1(v)⊂BR(x0) is (E,m)-NS, and set(50)q=e-γm,ll.Then(A)
the kernelΠψj(x,y0) of the spectral projection Πψj=ψjψj, considered as a function of x, is (l+1,q)-dominated in x∈BR(x0) and globally bounded by 1;(B)
if the ballBLk(u) is also E-NR, then the Green function (51)BRx0∋x⟼GBLkux,y;Eis(l+1,q)-dominated in x∈BR(x0) and globally bounded by eLβ.By Hermitian symmetry of the Green functions, the counterparts of assertions (A) and (B) relative to the second argument, y∈BR(y0), also hold true, with x=x0 fixed.In the next definition, we use a parameterϱ=(α-1)/2; with α=4/3, one obtains ϱ=1/6 and (1+ϱ)/α=7/8.Definition 11.
A ballBL(u) is called m-localized (m-Loc, in short) if every eigenfunction ψj of the operator HBL(u) satisfies(52)ψjxψjy≤e-γm,Lρx,yfor any points x,y∈BL(u) with ρ(x,y)≥L1+ϱ/α≡L7/8.Definition 12.
A pair of ballsBL(x), BL(y) is calleddistant iff ρ(x,y)≥CNL, with CN=2AN+3, AN=4N.(The role of the constantsAN and CN will become clear in Section 4.3.)Definition 13.
A ballBlα(u) is called m-tunneling (m-T) if it contains a pair of distant m-NLoc balls Bl(x), Bl(y) and m-nontunneling (m-NT), otherwise.It is to be emphasized that, unlike the property ofE-resonance or (E,m)-singularity, the tunneling propertyis not related to a specific value of energyE, and tunneling even ina single ball occurs with a small probability. This is an advantage of the DSA compared to the fixed-energy MSA; yet, as was said in Introduction, the DSA provides final decay bounds for the Green functions in an entire range of energies, like the variable-energy MSA. However, unlike the MSA, the DSA also provides bounds on the eigenbases in the finite balls.Lemma 14.
Let be givenm-localized ball B=BL(u). If, for some E∈R, B is also E-NR, then it is (E,m)-NS.Proof.
The matrix elements of the resolventGB(E) can be assessed as follows:(53)GBx,y;E≤∑Ej∈ΣHBψjxψjyE-Ej.Here Σ(HB) is the spectrum of the operator HB.
Ifdist(E,Σ(HB))≥e-Lβ and ln(2L+1)N≤Lβ, then the m-Loc property implies (54)GBx,y;E≤e-γm,LL+Lβ+lnB≤e-γm,LL+2Lβ.Observe that, withm≥1 and 1-τ>β, one has for L0 large enough (hence, Lk large enough, k≥0)(55)m1+Lk-τLk-2Lkβ=mLk+mLk1-τ-2Lkβ≥m1+12Lk-τLk.We will see that the condition m≥1 is fulfilled for |g| large enough (cf. Lemma 17).Notice that the above proof uses the following fact: the cardinality of anN-particle ball of radius L is polynomially bounded in L. This will be used in Section 8. The same remains true for the symmetric powers of graphs Z satisfying the growth condition (129); this includes all lattices Z=Zd, d≥1.Lemma 15.
There isL~(1)∈N such that for L0≥L~(1)(A)
if a ballBLk+1(u) is E-CNR and contains no pair of distant (E,m)-S balls BLk(v), BLk(w), then it is also (E,m)-NS;(B)
if a ballBLk+1(u) is m-NT and E-CNR, then it is also (E,m)-NS.Proof.
(A) By assumption, eitherBLk+1(u) is E-CNR and contains no (E,m)-S ball of radius Lk or there is a point w∈BLk+1(u) such that any ball BLk(v)⊂BLk(u) with ρ(w,v)≥CNLk is (E,m)-NS. We will obtain the required bound without using the balls with ρ(w,v)<CNLk.
In the former case, such an exclusion is unnecessary, but in order to treat both situations with one argument, we can formally setw=u (in fact, the choice of the center of a “fictitious” singular ball does not really matter).
Fix arbitrary pointsx, y with R≔ρ(x,y)>Lk1+ϱ; as before, ϱ=1/6. By the triangle inequality, (56)ρx,BCN-1Lkw+ρy,BCN-1Lkw≥R-2CN-2Lk. Assume first that (57)r′≔ρx,BCN-1Lkw≥Lk+1,r′′≔ρy,BCN-1Lkw≥Lk+1.Inside Br′(x)∪Br′′(y), all the balls of radius Lk are automatically disjoint from B2Lk(w); thus they are (E,m)-NS. Furthermore, (58)r′+r′′≥R-2CN-1Lk-2≥R-2CNLk.
Consider the setB=Br′(x)×Br′′(y) and the function f:B→C defined by f(x′,x′′)=GBLk+1(x′,x′′;E). Since E is not a pole of the resolvent GBLk+1(u)(·), its matrix elements are well defined, hence bounded on a finite set. By Lemma 10, f is (Lk,q)-dominated both in x′ and in x′′, with q≤e-γ(m,Lk,n). Therefore, one can write, with the convention -ln0=+∞, using Lemma 7,(59)-lnfu,y≥-lne-m1+1/2Lk-τLkR-2CNLk-2Lk/Lk+1eLk+1β=m1+12Lk-τLkLk+1R1-3CNR-1Lk-Lk+1β=mR1+12Lk-τ1-Lk-11-3CNLk-ϱ-Lk+1βmR≥m1+14Lk-τR≥γm,Lk+1R.If r′=0 (resp., r′′=0), the required bound follows from the dominated decay of the function f(x′,x′′) in x′′ (resp., in x′).
(B) Assume otherwise. Then, by assertion (A), theE-CNR ball BLk+1(u) must contain a pair of distant (E,m)-S balls BLk(x), BLk(y). Both of them are E-NR, since BLk+1(u) is E-CNR. By virtue of Lemma 14, both BLk(x) and BLk(y) must be m-NLoc, so that BLk+1(u) must be m-T, which contradicts the hypothesis.Lemma 16.
There isL~(2)∈N such that, for all L0≥L~(2), if for any E∈R a ball BLk+1(u) contains no pair of distant (E,m)-S balls of radius Lk, then it is m-Loc.Proof.
One can proceed as in the proof of the previous lemma, but with the functionsfj:(v′,v′′)↦|ψj(v′)ψj(v′′)|, where ψj,j∈[1,|BLk+1(u)|], are normalized eigenfunctions of operator HBLk+1(u). Notice that the Ej-nonresonance condition for BLk+1(u) is not required here, since one has ψj=1, so the function fj is globally bounded by 1. Let x,y∈BLk+1(u) and assume ρ(x,y)=R≥Lk1+ϱ.
Arguing as in the previous lemma and using the dominated decay of the functionfj(·,·) (cf. Lemma 10), we obtain, for any pair of points with ρ(y′,y′′)≥Lk1+ϱ,(60)-lnψjxψjy≥-lne-m1+1/2Lk-τLkR-2CNLk-2Lk/Lk+1.A direct comparison with the RHS of the first equation in (59) shows that the RHS of (60) is bigger, owing to the absence of the large factor eLkβ. Therefore, it admits the same (or better) lower bound as in (59).Below we summarize the relations between the parameters used in the above proof:(61)ρx,y≥Lk-11+ϱ,0<τ<ρ,1+ϱ<α,β<1-τ.These conditions are fulfilled, for example, with(62)α=43,ρ=16,τ=18,β=12.


## 3.1. Geometric Resolvent Inequality

The most essential part of the Direct Scaling Analysis concerns the finite-volume approximationsHΛ(ω) of the random Hamiltonian H(ω), acting in finite-dimensional spaces HΛ, with Λ⊂Z>N, |Λ|≡cardΛ<∞. We define (34)HΛω≔1ΛHω1Λ↾HΛ,where the indicator function 1Λ is identified with the operator of multiplication by 1Λ. HΛ is a bounded, finite-dimensional Hermitian operator, so its spectrum Σ(HΛ) is real.Operator(-Δ) can be represented as follows: (35)-Δ=-nZN+∑x,yΓxy+Γyx,Γxyfx≔δxyfy, and δxy is the Kronecker symbol.Recall that we denote byGΛ(E)=(HΛ-E)-1 the resolvent of HΛ and by G(x,y;E) the matrix elements thereof in the standard delta-basis (the Green functions).The so-called Geometric Resolvent Inequality (GRI) for the Green functions can be easily deduced from the second resolvent identity (we omitE for brevity):(36)GBLx,y≤Clmaxv∈∂-BlxGBlxx,vmaxv′∈∂+BlxGBLv′,y,where(37)Cl=∂Blv≤CNlN.See, for example, [22]. Clearly, (36) implies the inequality(38)GBLx,y≤Clmaxρx,v=lGBlxx,vmaxρu,v′≤l+1GBLv′,y.Sometimes we use another inequality stemming from (36):(39)GBLx,y;E≤ClGBluEmaxv:ρu,v≤l+1GBLv,y;E.Similarly, for the solutions ψ of the eigenfunction equation Hψ=Eψ one has, with x∈Bl(u),(40)ψx≤ClGBlxEmaxy:ρx,y≤l+1ψy,provided that E is not an eigenvalue of the operator HBl(u).


## 3.2. Dominated Decay Bounds for the Green Functions and Eigenfunctions

The analytic statements of this subsection apply indifferently to any discrete Schrödinger operators, on fairly general graphs, regardless of their single- or multiparticle structure of the potential energy. To avoid any confusion, in this subsection we denote the underlying graph byG; in applications to N-particle models in Z, one has to set G=Z>N.Definition 5.
Fix an integerl≥0 and a number q∈(0,1). Consider a finite connected subgraph Λ⊂G and a ball BR(u)⊊Λ. A function f:BR(u)→R+ is called (l,q)-dominated in BR(u) if for any ball Bl(x)⊂ΛR(u) one has(41)fx≤qmaxy:dx,y≤lfy.Lemma 6.
Let be given integersL≥l≥0 and a number q∈(0,1). If f:Λ→R+ on a finite connected graph Λ is (l,q)-dominated in a ball BL(x)⊊Λ, then(42)fx≤qL+1/l+1Mf,B≤qL-l/l+1Mf,B,Mf,Λ≔maxx∈Λfx.Proof.
The claim follows from (41) by induction (“radial descent”): (43)fx≤qMf,Bl+1≤⋯≤qjMf,Bjl+1≤⋯≤qL+1/l+1Mf,B.Lemma 7.
LetΛ be a finite connected graph and f:Λ×Λ→R+, (x′,x′′)↦f(x′,x′′), a function which is separately (l,q)-dominated in x′∈Br′(u′)⊊Λ and in x′′∈Br′′(u′′)⊊Λ, with d(u′,u′′)≥r′+r′′+2. Then(44)fu′,u′′≤qr′+1/l+1+r′′+1/l+1Mf,B≤qr′+r′′-2l/l+1Mf,B.Proof.
Fix any pointy′′∈Br′′+1(u′′) and consider the function (45)fy′′:y′⟼fy′,y′′ which is (l,q)-dominated in Br′(u′)⊊Λ, so by Lemma 6,(46)fy′′u′=fu′,y′′≤qr′+1/l+1Mf,Λ×Λ.Now introduce the function f~u′:y′′↦f(u′,y′′) which is (l,q)-dominated in Br′′(u′′)⊊Λ and bounded by the RHS of (46). Applying again Lemma 6, the claim follows.The relevance of the above notions and results is illustrated by Lemma10, stated in the next subsection. It could be stated in a general form, for arbitrary countable graphs, but its formulation relies on the notions of “nonsingular” and “nonresonant” balls. For our model, these are given in Definitions 8 and 9. However, the proof given in [14] (cf. [14, Lemma 4]) applies indifferently to arbitrary graphs.


## 3.3. Localization and Tunneling in Finite Balls

From this point on, we will work with a sequence of length scales, positive integers{Lk,k≥0} defined recursively by Lk+1=⌈Lkα⌉, L0>2. For clarity, we keep the value α=4/3; observe that α2<2. In several arguments we will require the initial scale L0 to be large enough.Definition 8.
Given a sample of the random potentialV(·;ω), a ball BL(u) is called (i)
E-nonresonant (E-NR) if GB(E;ω)≤e+Lβ and E-resonant (E-R) otherwise;(ii)
completelyE-nonresonant (E-CNR) if it does not contain any E-R ball Bl(u)⊆BL(u) with l≥L1/α and E-partially resonant (E-PR) otherwise.Definition 9.
Given a sampleV(·;ω), a ball BL(u) is called (E,m)-nonsingular ((E,m)-NS) if(47)maxy∈∂-BLuGBLuu,y;E;ω≤e-γm,LL+2Lβ,where(48)γm,L≔m1+L-τ,τ=18.Otherwise, it is called (E,m)-singular ((E,m)-S).Lemma 10 (cf. [14, Lemma 4]).
Consider a ballBLk(u) and an operator H=HBLk(u) with fixed (nonrandom) potential W. Let {ψj,j=1,…,|BLk(u)|} be the normalized eigenfunctions of H. Pick a pair of points x0,y0∈BLk(u) with d(x0,y0)>Lk-1+1 and an integer (49)R∈Lk-1,dx0,y0-Lk-1+1. Suppose that any ball BLk-1(v)⊂BR(x0) is (E,m)-NS, and set(50)q=e-γm,ll.Then(A)
the kernelΠψj(x,y0) of the spectral projection Πψj=ψjψj, considered as a function of x, is (l+1,q)-dominated in x∈BR(x0) and globally bounded by 1;(B)
if the ballBLk(u) is also E-NR, then the Green function (51)BRx0∋x⟼GBLkux,y;Eis(l+1,q)-dominated in x∈BR(x0) and globally bounded by eLβ.By Hermitian symmetry of the Green functions, the counterparts of assertions (A) and (B) relative to the second argument, y∈BR(y0), also hold true, with x=x0 fixed.In the next definition, we use a parameterϱ=(α-1)/2; with α=4/3, one obtains ϱ=1/6 and (1+ϱ)/α=7/8.Definition 11.
A ballBL(u) is called m-localized (m-Loc, in short) if every eigenfunction ψj of the operator HBL(u) satisfies(52)ψjxψjy≤e-γm,Lρx,yfor any points x,y∈BL(u) with ρ(x,y)≥L1+ϱ/α≡L7/8.Definition 12.
A pair of ballsBL(x), BL(y) is calleddistant iff ρ(x,y)≥CNL, with CN=2AN+3, AN=4N.(The role of the constantsAN and CN will become clear in Section 4.3.)Definition 13.
A ballBlα(u) is called m-tunneling (m-T) if it contains a pair of distant m-NLoc balls Bl(x), Bl(y) and m-nontunneling (m-NT), otherwise.It is to be emphasized that, unlike the property ofE-resonance or (E,m)-singularity, the tunneling propertyis not related to a specific value of energyE, and tunneling even ina single ball occurs with a small probability. This is an advantage of the DSA compared to the fixed-energy MSA; yet, as was said in Introduction, the DSA provides final decay bounds for the Green functions in an entire range of energies, like the variable-energy MSA. However, unlike the MSA, the DSA also provides bounds on the eigenbases in the finite balls.Lemma 14.
Let be givenm-localized ball B=BL(u). If, for some E∈R, B is also E-NR, then it is (E,m)-NS.Proof.
The matrix elements of the resolventGB(E) can be assessed as follows:(53)GBx,y;E≤∑Ej∈ΣHBψjxψjyE-Ej.Here Σ(HB) is the spectrum of the operator HB.
Ifdist(E,Σ(HB))≥e-Lβ and ln(2L+1)N≤Lβ, then the m-Loc property implies (54)GBx,y;E≤e-γm,LL+Lβ+lnB≤e-γm,LL+2Lβ.Observe that, withm≥1 and 1-τ>β, one has for L0 large enough (hence, Lk large enough, k≥0)(55)m1+Lk-τLk-2Lkβ=mLk+mLk1-τ-2Lkβ≥m1+12Lk-τLk.We will see that the condition m≥1 is fulfilled for |g| large enough (cf. Lemma 17).Notice that the above proof uses the following fact: the cardinality of anN-particle ball of radius L is polynomially bounded in L. This will be used in Section 8. The same remains true for the symmetric powers of graphs Z satisfying the growth condition (129); this includes all lattices Z=Zd, d≥1.Lemma 15.
There isL~(1)∈N such that for L0≥L~(1)(A)
if a ballBLk+1(u) is E-CNR and contains no pair of distant (E,m)-S balls BLk(v), BLk(w), then it is also (E,m)-NS;(B)
if a ballBLk+1(u) is m-NT and E-CNR, then it is also (E,m)-NS.Proof.
(A) By assumption, eitherBLk+1(u) is E-CNR and contains no (E,m)-S ball of radius Lk or there is a point w∈BLk+1(u) such that any ball BLk(v)⊂BLk(u) with ρ(w,v)≥CNLk is (E,m)-NS. We will obtain the required bound without using the balls with ρ(w,v)<CNLk.
In the former case, such an exclusion is unnecessary, but in order to treat both situations with one argument, we can formally setw=u (in fact, the choice of the center of a “fictitious” singular ball does not really matter).
Fix arbitrary pointsx, y with R≔ρ(x,y)>Lk1+ϱ; as before, ϱ=1/6. By the triangle inequality, (56)ρx,BCN-1Lkw+ρy,BCN-1Lkw≥R-2CN-2Lk. Assume first that (57)r′≔ρx,BCN-1Lkw≥Lk+1,r′′≔ρy,BCN-1Lkw≥Lk+1.Inside Br′(x)∪Br′′(y), all the balls of radius Lk are automatically disjoint from B2Lk(w); thus they are (E,m)-NS. Furthermore, (58)r′+r′′≥R-2CN-1Lk-2≥R-2CNLk.
Consider the setB=Br′(x)×Br′′(y) and the function f:B→C defined by f(x′,x′′)=GBLk+1(x′,x′′;E). Since E is not a pole of the resolvent GBLk+1(u)(·), its matrix elements are well defined, hence bounded on a finite set. By Lemma 10, f is (Lk,q)-dominated both in x′ and in x′′, with q≤e-γ(m,Lk,n). Therefore, one can write, with the convention -ln0=+∞, using Lemma 7,(59)-lnfu,y≥-lne-m1+1/2Lk-τLkR-2CNLk-2Lk/Lk+1eLk+1β=m1+12Lk-τLkLk+1R1-3CNR-1Lk-Lk+1β=mR1+12Lk-τ1-Lk-11-3CNLk-ϱ-Lk+1βmR≥m1+14Lk-τR≥γm,Lk+1R.If r′=0 (resp., r′′=0), the required bound follows from the dominated decay of the function f(x′,x′′) in x′′ (resp., in x′).
(B) Assume otherwise. Then, by assertion (A), theE-CNR ball BLk+1(u) must contain a pair of distant (E,m)-S balls BLk(x), BLk(y). Both of them are E-NR, since BLk+1(u) is E-CNR. By virtue of Lemma 14, both BLk(x) and BLk(y) must be m-NLoc, so that BLk+1(u) must be m-T, which contradicts the hypothesis.Lemma 16.
There isL~(2)∈N such that, for all L0≥L~(2), if for any E∈R a ball BLk+1(u) contains no pair of distant (E,m)-S balls of radius Lk, then it is m-Loc.Proof.
One can proceed as in the proof of the previous lemma, but with the functionsfj:(v′,v′′)↦|ψj(v′)ψj(v′′)|, where ψj,j∈[1,|BLk+1(u)|], are normalized eigenfunctions of operator HBLk+1(u). Notice that the Ej-nonresonance condition for BLk+1(u) is not required here, since one has ψj=1, so the function fj is globally bounded by 1. Let x,y∈BLk+1(u) and assume ρ(x,y)=R≥Lk1+ϱ.
Arguing as in the previous lemma and using the dominated decay of the functionfj(·,·) (cf. Lemma 10), we obtain, for any pair of points with ρ(y′,y′′)≥Lk1+ϱ,(60)-lnψjxψjy≥-lne-m1+1/2Lk-τLkR-2CNLk-2Lk/Lk+1.A direct comparison with the RHS of the first equation in (59) shows that the RHS of (60) is bigger, owing to the absence of the large factor eLkβ. Therefore, it admits the same (or better) lower bound as in (59).Below we summarize the relations between the parameters used in the above proof:(61)ρx,y≥Lk-11+ϱ,0<τ<ρ,1+ϱ<α,β<1-τ.These conditions are fulfilled, for example, with(62)α=43,ρ=16,τ=18,β=12.


## 4. Scaling Analysis of Eigenfunctions: Finite-Range Interaction

### 4.1. Initial Scale Estimates

Lemma 17 (initial scale bound).
For anyL0>2, m≥1, and p>0 there is g0=g0(m,p,L0)<∞ such that for all g with |g|≥g0, any ball BL0(u), and any E∈R(63)PBL0uisE,m-S≤L0-p,PBL0uism-NLoc≤L0-p.See the proof in the Appendix.


### 4.2. EVC Bounds for Distant Pairs of Balls

The following statement reflects the progress achieved in the area of multiparticle eigenvalue concentration (EVC) estimates since the time when manuscript [18] had appeared.Theorem 18 (cf. [21, Theorem 4], [19, Lemma 2.4]).
LetV:Z×Ω→R be a random field satisfying (W3) with parameters b,b′∈(0,1], A′∈(0,+∞). For all L>0 large enough, any distant pair of balls BLN(u′), BLN(u′′) the following bound holds for any s∈(0,1]:(64)PdistΣL,u′,ΣL,u′′≤s≤3L2NhL2s,where ΣL,u′, ΣL,u′′ are the respective spectra of HBL(u′) and HBL(u′′), and(65)hLs≔C′′LA′′sb∧b′,b∧b′≡minb,b′,and A′′=A′′(A′,N)∈(0,+∞). Consequently, this implies the following bound for β′∈(0,β), all L large enough and any pair u′, u′′, with d(u′,u′′)>4NL: (66)P∃E∈R:BLu′,BLu′′areE-PR≤e-Lβ′.A considerable advantage of the above EVC bound, compared to [4, Lemma 2], is that it holds for any distant pair of balls. In [4], as well as in [17], a similar bound is established only for pairs of balls sufficiently distant with respect to the so-called Hausdorff (pseudo)metric; such a bound did not cover some pairs distant with respect to the more natural symmetrized distance, like ρ. As a result, one could not prove uniform decay of eigenfunction correlators in the physically relevant distance in the multiparticle configuration space. A similar difficulty was encountered in [6].


### 4.3. Partially InteractiveN-Particle Balls

Definition 19.
AnN-particle ball BL(u) is called partially interactive (PI, in short) if diamu>ANL, AN=4N, and fully interactive (FI) otherwise.The local HamiltonianHBL(u)(N) with a finite-range interaction in a PI ball is algebraically decomposable as follows:(67)HBLuN=HBLu′n′⊗1n′′+1n′⊗HBLu′′n′′with n′+n′′=N, u=(u′,u′′), u′∈Z>n′, u′′∈Z>n′′, while for a fully interactive ball one cannot guarantee such a property. (In the case of a rapidly decaying interaction, the RHS is close in norm to the LHS.)Lemma 20.
(A) If BL(x) is PI, then there exists a partition of [[1,N]] into nonempty complementary subsets J, Jc such that ρ(ΠJBL(x),ΠJcBL(x))>2L.
(B) If BL(x), BL(y) are two FI balls with ρ(x,y)>11NL, then ρ(ΠBL(x),ΠBL(y))≥2L.Here the value11N is obtained as 2AN+3.See the proof in the Appendix, SectionA.2.Now we state a result on localization in PI balls for the Hamiltonians with a short-range interaction (condition (U0)). The reader may want to compare the elementary proof of Lemma 21 below with that of Lemma  3 in [4] which is substantially more complex. The simplification comes here from the fact that we rely on the m-Loc property, that is, exponential decay of all EFs of a Hamiltonian in a given ball, and this property is nonrelated to a specific energy but covers the entire spectrum of the Hamiltonian at hand; this results in a few-line proof of Lemma 21 as well as of an important probabilistic bound (Lemma 22) stemming from it.Lemma 21.
Assume that the interactionU has finite range r0∈[0,+∞) (cf. (U0)). Consider a PI N-particle ball with canonical factorization (68)BLkNu=BLkn′u′×BLkn′′u′′. If the balls BLk-1n′(u′) and BLk-1n′′(u′′) are m-Loc, then the ball BLk(N)(u) is also m-Loc.Proof.
The interaction between subsystems corresponding tou′ and u′′ vanishes, so HBLk(u) admits the decomposition(69)HBLku=HBLkn′u′⊗1n′′+1n′⊗HBLkn′′u′′.Thus the eigenfunctions Ψj of HBLk(u) can be chosen in the form Ψj=ϕj′⊗ψj′′, where ϕj′ are eigenfunctions of HBLkn′(u′) and, respectively, ψj′′ are eigenfunctions of HBLkn′′(u′′). Now the required decay bounds on the projection kernel, (70)ΨjxΨjy=ϕj′x′ϕj′′x′′ψj′y′ψj′′y′′,follow directly from the respective bounds on ϕ and ψ, which are guaranteed by the m-Loc hypotheses on the projection balls BLkn′(u′) and BLkn′′(u′′).Given numbersp,θ>0, introduce the quantities(71)Pn,k=2N-np1+θk,1≤n≤N;k≥0.Later it will be required that ϑ∈(0,2-1).Lemma 22.
Suppose that for alln∈[1,N-1] and for a given k≥0 the following bound holds for any n-particle ball BLk(n)(u): (72)PBLknuism-NLoc≤Lk-Pn,k. Then for any N-particle PI ball BLk(N)(u) one has (73)PBLkNuism-NLoc≤2Lk-2PN,k≤14Lk-PN,k.Proof.
It follows from Lemma21 that a ball BLk(N)(u) admitting a factorization of the form (69) is m-NLoc only if at least one of the projection balls BLkn′(u′), BLkn′′(u′′) is m-NLoc. Therefore, with L0 large enough, we obtain (74)PBLkNuism-NLoc≤PBLkn′′u′′ism-NLoc+PBLkn′′u′′ism-NLoc≤Lk-Pn′,k+Lk-Pn′′,k≤2Lk-2PN,k<14Lk-PN,k.


### 4.4. Fully InteractiveN-Particle Balls

Owing to Lemma22, it remains to establish localization bounds for N-particle FI balls BLk+1(x), assuming, if necessary, similar bounds:(i)
Forn-particle balls of any radius Lk′, k′≥0, with n<N.(ii)
ForN-particle balls of any radius Lk′ with k′≤k.Lemma 23 (main inductive lemma).
Suppose that for a givenk≥0 and all i∈[0,k] the following bound holds for any N-particle ball BLk(N)(u):(75)PBLiNuism-NLoc≤Li-PN,i.Then for any N-particle FI ball BLk+1(n)(x) one has(76)PBLk+1Nuism-NLoc≤Lk+1-PN,k+1,provided that(77)p>2α22-α2Nd,0<3θ≤min2-α2α2-2Ndp,2-1.Furthermore, for any pair of distant balls BLk(N)(u), BLk(N)(v), one has, for some a,c>0,(78)P∃E∈R:BLkNu,BLkNvareE,m-S≤Lk-PN,k≡Lk-p1+ϑk≤e-aln1+cLk.Proof.
Introduce the following events:(79)Nk+1=BLk+1Nuism-NLoc,Sk2=∃E∃distantE,m-SballsBLkNu,BLkNv⊂BLk+1Nw,Rk2=∃E∃distantE-PRballsBLkNu,BLkNv⊂BLk+1Nw. By virtue of Lemma 16, we have (80)Nk+1⊂Sk2⊂Rk2∪Sk2∖Rk2, and by Theorem 18, PRk(2)≤e-Lkβ′, so it remains to assess the probability PSk(2)∖Rk(2). Fix points u,v∈BLk+1(N)(w) and introduce the event (figuring in (78)) (81)Sk2u,v=∃E:BLkNu,BLkNvareE,m-S.Within the event Sk(2)(u,v)∖Rk(2), either BLk(N)(u) or BLk(N)(v) must be E-CNR; w.l.o.g., assume that BLk(N)(u) is E-CNR. Since it is (E,m)-S, Lemma 15 implies that it must contain a pair of m-NLoc balls BLk-1N(y′), BLk-1N(y′′) with d(y′,y′′)>1/2Lk-11+δ. Consider two possible situations:
(1) Both BLk-1N(y′) and BLk-1N(y′′) are FI.
Apply Lemma20, assertion (B): (82)ρΠBLk-1u,ΠBLk-1v≥2Lk. Using the inductive assumption (75) and the mixing condition (W2), we have, with Lk-1=Lk+11/α2 and L0 large enough, (83)PBLk-1Ny′,BLk-1Ny′′arem-NLoc≤2Lk+1-2p/α21+θk-1.The number of such pairs y′, y′′ is bounded by 1/2|BLk(N)(w)|2≤32N/2Lk2N, so in this case,(84)PSk2∖Rk2≤32N2Lk+1-2p/α21+θk-1+2N.A straightforward calculation shows that, for L0 large enough, the RHS is bounded by 1/4Lk+1-P(N,k+1), under conditions (77). Indeed, we need the inequality(85)2pα22N-Np1+θk-1-2Nd>2N-Np1+θk+1or, equivalently,(86)1+θ2<2pα2-2Ndp1+θk-1. By construction, ϑ∈(0,1), so one has (1+θ)2<1+3ϑ, and 2Nd/p>2Nd/p(1+θ)k-1. Therefore, if p>2α2/2-α2N and (87)3ϑ≤2-α2α2-2Np, then (85) is also satisfied, so that(88)PSk2≤PRk2+PSk2∖Rk2≤e-Lkβ′+14Lk+1-PN,k+1<12Lk+1-PN,k+1,yielding assertion (76) in case (1).
(2) Either BLk-1N(y′) or BLk-1N(y′′) is PI. Then by Lemma 22, (89)PBLk-1Ny′,BLk-1Ny′′ism-NLoc≤minPBLk-1Ny′ism-NLoc,PBLk-1Ny′′ism-NLoc≤Lk-1-2p1+θk-1<Lk+1-p1+θk+1, provided that (1+θ)2<2; that is, θ<2-1.
Finally, a straightforward calculation shows that for somea,c>0 one has the inequality (90)Lk-PN,k≡Lk-p1+ϑk≤e-aln1+cLk. This completes the proof.Applying Lemmas17 and 22, we come by induction to the following result.Theorem 24 (localization at any scale).
Assume that|g| is large enough, so that (63) hold true with p>2α2/2-α2Nd. Then for any k≥0 and any u∈ZNd(91)PBLkNuism-NLoc≤14Lk-p1+θkwith θ>0 satisfying (77). In other words, with probability ≥1-1/4Lk-p(1+θ)k, for every eigenfunction Ψj of operator HBLk(N)(u) and for all x,y∈BLk(N)(u) such that x-y≥Lk1+δ+ρ/α one has (92)ΨjxΨjy≤e-γm,LkLk<e-mLk.Theorem24 marks the end of the Direct Scaling Analysis of localized eigenfunctions in arbitrarily large finite balls.


## 4.1. Initial Scale Estimates

Lemma 17 (initial scale bound).
For anyL0>2, m≥1, and p>0 there is g0=g0(m,p,L0)<∞ such that for all g with |g|≥g0, any ball BL0(u), and any E∈R(63)PBL0uisE,m-S≤L0-p,PBL0uism-NLoc≤L0-p.See the proof in the Appendix.


## 4.2. EVC Bounds for Distant Pairs of Balls

The following statement reflects the progress achieved in the area of multiparticle eigenvalue concentration (EVC) estimates since the time when manuscript [18] had appeared.Theorem 18 (cf. [21, Theorem 4], [19, Lemma 2.4]).
LetV:Z×Ω→R be a random field satisfying (W3) with parameters b,b′∈(0,1], A′∈(0,+∞). For all L>0 large enough, any distant pair of balls BLN(u′), BLN(u′′) the following bound holds for any s∈(0,1]:(64)PdistΣL,u′,ΣL,u′′≤s≤3L2NhL2s,where ΣL,u′, ΣL,u′′ are the respective spectra of HBL(u′) and HBL(u′′), and(65)hLs≔C′′LA′′sb∧b′,b∧b′≡minb,b′,and A′′=A′′(A′,N)∈(0,+∞). Consequently, this implies the following bound for β′∈(0,β), all L large enough and any pair u′, u′′, with d(u′,u′′)>4NL: (66)P∃E∈R:BLu′,BLu′′areE-PR≤e-Lβ′.A considerable advantage of the above EVC bound, compared to [4, Lemma 2], is that it holds for any distant pair of balls. In [4], as well as in [17], a similar bound is established only for pairs of balls sufficiently distant with respect to the so-called Hausdorff (pseudo)metric; such a bound did not cover some pairs distant with respect to the more natural symmetrized distance, like ρ. As a result, one could not prove uniform decay of eigenfunction correlators in the physically relevant distance in the multiparticle configuration space. A similar difficulty was encountered in [6].


## 4.3. Partially InteractiveN-Particle Balls

Definition 19.
AnN-particle ball BL(u) is called partially interactive (PI, in short) if diamu>ANL, AN=4N, and fully interactive (FI) otherwise.The local HamiltonianHBL(u)(N) with a finite-range interaction in a PI ball is algebraically decomposable as follows:(67)HBLuN=HBLu′n′⊗1n′′+1n′⊗HBLu′′n′′with n′+n′′=N, u=(u′,u′′), u′∈Z>n′, u′′∈Z>n′′, while for a fully interactive ball one cannot guarantee such a property. (In the case of a rapidly decaying interaction, the RHS is close in norm to the LHS.)Lemma 20.
(A) If BL(x) is PI, then there exists a partition of [[1,N]] into nonempty complementary subsets J, Jc such that ρ(ΠJBL(x),ΠJcBL(x))>2L.
(B) If BL(x), BL(y) are two FI balls with ρ(x,y)>11NL, then ρ(ΠBL(x),ΠBL(y))≥2L.Here the value11N is obtained as 2AN+3.See the proof in the Appendix, SectionA.2.Now we state a result on localization in PI balls for the Hamiltonians with a short-range interaction (condition (U0)). The reader may want to compare the elementary proof of Lemma 21 below with that of Lemma  3 in [4] which is substantially more complex. The simplification comes here from the fact that we rely on the m-Loc property, that is, exponential decay of all EFs of a Hamiltonian in a given ball, and this property is nonrelated to a specific energy but covers the entire spectrum of the Hamiltonian at hand; this results in a few-line proof of Lemma 21 as well as of an important probabilistic bound (Lemma 22) stemming from it.Lemma 21.
Assume that the interactionU has finite range r0∈[0,+∞) (cf. (U0)). Consider a PI N-particle ball with canonical factorization (68)BLkNu=BLkn′u′×BLkn′′u′′. If the balls BLk-1n′(u′) and BLk-1n′′(u′′) are m-Loc, then the ball BLk(N)(u) is also m-Loc.Proof.
The interaction between subsystems corresponding tou′ and u′′ vanishes, so HBLk(u) admits the decomposition(69)HBLku=HBLkn′u′⊗1n′′+1n′⊗HBLkn′′u′′.Thus the eigenfunctions Ψj of HBLk(u) can be chosen in the form Ψj=ϕj′⊗ψj′′, where ϕj′ are eigenfunctions of HBLkn′(u′) and, respectively, ψj′′ are eigenfunctions of HBLkn′′(u′′). Now the required decay bounds on the projection kernel, (70)ΨjxΨjy=ϕj′x′ϕj′′x′′ψj′y′ψj′′y′′,follow directly from the respective bounds on ϕ and ψ, which are guaranteed by the m-Loc hypotheses on the projection balls BLkn′(u′) and BLkn′′(u′′).Given numbersp,θ>0, introduce the quantities(71)Pn,k=2N-np1+θk,1≤n≤N;k≥0.Later it will be required that ϑ∈(0,2-1).Lemma 22.
Suppose that for alln∈[1,N-1] and for a given k≥0 the following bound holds for any n-particle ball BLk(n)(u): (72)PBLknuism-NLoc≤Lk-Pn,k. Then for any N-particle PI ball BLk(N)(u) one has (73)PBLkNuism-NLoc≤2Lk-2PN,k≤14Lk-PN,k.Proof.
It follows from Lemma21 that a ball BLk(N)(u) admitting a factorization of the form (69) is m-NLoc only if at least one of the projection balls BLkn′(u′), BLkn′′(u′′) is m-NLoc. Therefore, with L0 large enough, we obtain (74)PBLkNuism-NLoc≤PBLkn′′u′′ism-NLoc+PBLkn′′u′′ism-NLoc≤Lk-Pn′,k+Lk-Pn′′,k≤2Lk-2PN,k<14Lk-PN,k.


## 4.4. Fully InteractiveN-Particle Balls

Owing to Lemma22, it remains to establish localization bounds for N-particle FI balls BLk+1(x), assuming, if necessary, similar bounds:(i)
Forn-particle balls of any radius Lk′, k′≥0, with n<N.(ii)
ForN-particle balls of any radius Lk′ with k′≤k.Lemma 23 (main inductive lemma).
Suppose that for a givenk≥0 and all i∈[0,k] the following bound holds for any N-particle ball BLk(N)(u):(75)PBLiNuism-NLoc≤Li-PN,i.Then for any N-particle FI ball BLk+1(n)(x) one has(76)PBLk+1Nuism-NLoc≤Lk+1-PN,k+1,provided that(77)p>2α22-α2Nd,0<3θ≤min2-α2α2-2Ndp,2-1.Furthermore, for any pair of distant balls BLk(N)(u), BLk(N)(v), one has, for some a,c>0,(78)P∃E∈R:BLkNu,BLkNvareE,m-S≤Lk-PN,k≡Lk-p1+ϑk≤e-aln1+cLk.Proof.
Introduce the following events:(79)Nk+1=BLk+1Nuism-NLoc,Sk2=∃E∃distantE,m-SballsBLkNu,BLkNv⊂BLk+1Nw,Rk2=∃E∃distantE-PRballsBLkNu,BLkNv⊂BLk+1Nw. By virtue of Lemma 16, we have (80)Nk+1⊂Sk2⊂Rk2∪Sk2∖Rk2, and by Theorem 18, PRk(2)≤e-Lkβ′, so it remains to assess the probability PSk(2)∖Rk(2). Fix points u,v∈BLk+1(N)(w) and introduce the event (figuring in (78)) (81)Sk2u,v=∃E:BLkNu,BLkNvareE,m-S.Within the event Sk(2)(u,v)∖Rk(2), either BLk(N)(u) or BLk(N)(v) must be E-CNR; w.l.o.g., assume that BLk(N)(u) is E-CNR. Since it is (E,m)-S, Lemma 15 implies that it must contain a pair of m-NLoc balls BLk-1N(y′), BLk-1N(y′′) with d(y′,y′′)>1/2Lk-11+δ. Consider two possible situations:
(1) Both BLk-1N(y′) and BLk-1N(y′′) are FI.
Apply Lemma20, assertion (B): (82)ρΠBLk-1u,ΠBLk-1v≥2Lk. Using the inductive assumption (75) and the mixing condition (W2), we have, with Lk-1=Lk+11/α2 and L0 large enough, (83)PBLk-1Ny′,BLk-1Ny′′arem-NLoc≤2Lk+1-2p/α21+θk-1.The number of such pairs y′, y′′ is bounded by 1/2|BLk(N)(w)|2≤32N/2Lk2N, so in this case,(84)PSk2∖Rk2≤32N2Lk+1-2p/α21+θk-1+2N.A straightforward calculation shows that, for L0 large enough, the RHS is bounded by 1/4Lk+1-P(N,k+1), under conditions (77). Indeed, we need the inequality(85)2pα22N-Np1+θk-1-2Nd>2N-Np1+θk+1or, equivalently,(86)1+θ2<2pα2-2Ndp1+θk-1. By construction, ϑ∈(0,1), so one has (1+θ)2<1+3ϑ, and 2Nd/p>2Nd/p(1+θ)k-1. Therefore, if p>2α2/2-α2N and (87)3ϑ≤2-α2α2-2Np, then (85) is also satisfied, so that(88)PSk2≤PRk2+PSk2∖Rk2≤e-Lkβ′+14Lk+1-PN,k+1<12Lk+1-PN,k+1,yielding assertion (76) in case (1).
(2) Either BLk-1N(y′) or BLk-1N(y′′) is PI. Then by Lemma 22, (89)PBLk-1Ny′,BLk-1Ny′′ism-NLoc≤minPBLk-1Ny′ism-NLoc,PBLk-1Ny′′ism-NLoc≤Lk-1-2p1+θk-1<Lk+1-p1+θk+1, provided that (1+θ)2<2; that is, θ<2-1.
Finally, a straightforward calculation shows that for somea,c>0 one has the inequality (90)Lk-PN,k≡Lk-p1+ϑk≤e-aln1+cLk. This completes the proof.Applying Lemmas17 and 22, we come by induction to the following result.Theorem 24 (localization at any scale).
Assume that|g| is large enough, so that (63) hold true with p>2α2/2-α2Nd. Then for any k≥0 and any u∈ZNd(91)PBLkNuism-NLoc≤14Lk-p1+θkwith θ>0 satisfying (77). In other words, with probability ≥1-1/4Lk-p(1+θ)k, for every eigenfunction Ψj of operator HBLk(N)(u) and for all x,y∈BLk(N)(u) such that x-y≥Lk1+δ+ρ/α one has (92)ΨjxΨjy≤e-γm,LkLk<e-mLk.Theorem24 marks the end of the Direct Scaling Analysis of localized eigenfunctions in arbitrarily large finite balls.


## 5. Exponential Decay of Eigenfunctions

The DSA provides exponential decay bounds on the eigenbases in finite balls, not just on the Green functions, which shows that quantum transport in an arbitrarily large finite system over distances comparable with the size of the system is strongly inhibited. The proof of exponential decay of EFs in an infinitely extended configuration space requires an additional argument, originally proposed by Fröhlich et al. [9] and reformulated by von Dreifus and Klein [10]. More than 20 years after [10], it is merely a simple exercise.Theorem 25.
ForP-a.e. ω∈Ω every normalized eigenfunction Ψ of operator H(ω) satisfies the following bound: for some R(ω), x^(ω), and all y with y≥R(ω)(93)Ψy≤e-my.Proof.
Pick an arbitrary vertexz∈Z>N. By the Borel-Cantelli lemma combined with the probabilistic bound from Lemma 23, there is a subset Ω′⊂Ω with PΩ′=1 such that for any ω∈Ω′ and some k0(ω), all k≥k0, and any E∈R there is no pair of Lk1+δ-distant (E,m)-S balls BLk(x),BLk(y)⊂BLk+2(z). Fix ω∈Ω′.
LetΨ be a normalized eigenfunction of H(ω) with eigenvalue λ. Since Ψ2≤1 implies Ψ∞≤1, there is a point x^ such that Ψ∞=|Ψ(x^)|. If x^n∈BLk-1(0), then BLk(0) must be (λ,m)-S; otherwise the (λ,m)-NS property would lead to a contradiction: (94)Ψ∞=Ψx^≤e-mLkΨ∞<Ψ∞. Thus any ball BLk(y)⊂BLk+2(z) with ρ(y,z)∈[Lk+1,Lk+2) (hence, with ρ(y,z)≥Lkα>2Lk1+δ) is (λ,m)-NS. Note that the function x↦|Ψn(x)| is (Lk,q)-dominated in BR(y), with R=ρ(y,z)-2Lk1+δ-1>1/2Lk1+2δ and q=e-γ(m,Lk)Lk. Recall that τ=δ/4 (cf. (62)). By Lemma 6, for Lk large enough, (95)-lnΨyρy,z≥m1+Lk-τ1-2Lk+1ρy,z≥m1+12Lk-τ>m, yielding assertion (93).


## 6. EF Correlators and Dynamical Localization

### 6.1. Bounds on the EFC in Finite Volumes

We use here a finite-volume variant of a “soft” argument proposed by Germinet and Klein in [11] and adapted in our earlier papers to finite-volume Hamiltonians, both single-particle (cf. [14, Memma 9]) and N-particle (cf. [19, Lemma 3.8]). Working with finite volumes allows us to avoid a functional-analytic complement regarding the weighted Hilbert-Schmidt norms of spectral projections of operators H(ω) in the entire graph Z>N and replace it with a simple application of Bessel’s inequality.Denote byB1(I) the set of all Borel functions ϕ:R→C with suppϕ⊂I and ϕ∞≤1.Proposition 26 (cf. [14, Lemma 9], [19, Lemma 3.8]).
Fix an integerL∈N∗ and assume that the following bound holds for a given pair of disjoint balls BL(x), BL(y) and an interval I⊆R:(96)P∃E∈I:BLx,BLyareE,m-S≤fL,for some f(L). Then for any connected subgraph Λ containing the union BL(x)∪BL(y), one has(97)Esupϕ∈B1I1xϕHΛω1y≤CLde-mL+fL.Taking into account Lemma23 establishing the probabilistic bound for the quantity f(L) figuring in the RHS of (96) and (97), we come to the following corollary of Proposition 26.Corollary 27.
Under the hypotheses of Theorem2, for any x,y∈Z>N with d(x,y)>2L+1, any connected subset Λ⊃BL(x)∪BL(y)(98)Esupϕ∈B1R1xϕHΛω1y≤Ce-aln1+cρx,y.


### 6.2. Dynamical Localization in an Infinite Graph

Here we follow the same path as in earlier works by Aizenman et al. (cf., e.g., [13, 23]).Recall that if the random potentialV is a.s. bounded, the norm of H(ω) is a.s. bounded by a constant, so its spectrum is contained in some nonrandom bounded interval I. Given an interval I⊂R, denote by C1(I) the set of all continuous functions ϕ on R with suppϕ⊂I and ϕ∞≤1.Theorem 28.
Consider the HamiltonianH(ω) of form (1) with a bounded random potential satisfying assumptions (W1)–(W3) and the interaction potential satisfying one of the assumptions (U0) and (U1). There is g0<+∞ such that if |g|≥g0, then the following property holds true.
For any bounded intervalI⊂R containing the a.s. spectrum of H(ω) and for all x,y∈Z>N, x≠y, and with the same c,a>0 as in (98),(99)Esupϕ∈C1I1xϕHω1y≤Ce-aln1+cρx,y.Proof.
We prove the claim under assumption (U0), using Corollary (97) established under this assumption. Similarly, case (U1) will follow from the analogous results proved in Section 7 for the infinite-range interactions.
Since bothV are U are bounded, the norm of the random operator H(ω) is a.s. bounded by a nonrandom constant; hence there exists a nonrandom bounded interval I⊂R containing a.s. the spectrum Σ(H(ω)). For any ball B and any points x,y∈B introduce a spectral measure μB,ωx,y uniquely defined by (100)∫ϕλdμB,ωx,yλ=1xϕHBωΠIHBω1y,where ϕ is an arbitrary bounded continuous function with suppϕ⊂I and similar spectral measures μωx,y for the operator H(ω) on the entire graph Z>N. If {BLk} is a growing sequence of balls, then μBLk,ωx,y converge vaguely to μωx,y as k→∞. Note that this fact holds true in a much more general context of unbounded operators: by a well-known result (cf. [24]), for the strong resolvent convergence of operators Hn→H, with a common core D, it suffices that Hnϕ→Hϕ for any element ϕ∈D; in turn, this implies the vague convergence of the spectral measures 〈φ,ϕ(Hn)ψ〉. In our case, we can choose as a core the subspace of all compactly supported functions, and on such functions, the operators Hn converge by stabilization.
So, by Fatou’s lemma on convergent measures, for any measurable setE⊂R and any growing sequence of balls BLk,(101)Eμωx,yE≤liminfk→∞EμBLkx,yE.Therefore, the uniform bounds in finite volumes BLk, established in Corollary 27, remain valid in the entire infinite graph.Now assertion (B) of the main Theorem 2 for finite-range interactions follows by taking continuous functions of the form (102)ϕt:λ⟼fIλeitλ,t∈R,where fI:R→[0,1] is an arbitrary continuous function interpolating between 1 on an interval I=[-R,R] containing a.s. the spectrum of H(ω) and 0 on R∖[-R-1,R+1].Note also that, arguing as in [23, Sect. 2.5], one can extend bound (99) to the compactly supported bounded Borel functions ϕ.


## 6.1. Bounds on the EFC in Finite Volumes

We use here a finite-volume variant of a “soft” argument proposed by Germinet and Klein in [11] and adapted in our earlier papers to finite-volume Hamiltonians, both single-particle (cf. [14, Memma 9]) and N-particle (cf. [19, Lemma 3.8]). Working with finite volumes allows us to avoid a functional-analytic complement regarding the weighted Hilbert-Schmidt norms of spectral projections of operators H(ω) in the entire graph Z>N and replace it with a simple application of Bessel’s inequality.Denote byB1(I) the set of all Borel functions ϕ:R→C with suppϕ⊂I and ϕ∞≤1.Proposition 26 (cf. [14, Lemma 9], [19, Lemma 3.8]).
Fix an integerL∈N∗ and assume that the following bound holds for a given pair of disjoint balls BL(x), BL(y) and an interval I⊆R:(96)P∃E∈I:BLx,BLyareE,m-S≤fL,for some f(L). Then for any connected subgraph Λ containing the union BL(x)∪BL(y), one has(97)Esupϕ∈B1I1xϕHΛω1y≤CLde-mL+fL.Taking into account Lemma23 establishing the probabilistic bound for the quantity f(L) figuring in the RHS of (96) and (97), we come to the following corollary of Proposition 26.Corollary 27.
Under the hypotheses of Theorem2, for any x,y∈Z>N with d(x,y)>2L+1, any connected subset Λ⊃BL(x)∪BL(y)(98)Esupϕ∈B1R1xϕHΛω1y≤Ce-aln1+cρx,y.


## 6.2. Dynamical Localization in an Infinite Graph

Here we follow the same path as in earlier works by Aizenman et al. (cf., e.g., [13, 23]).Recall that if the random potentialV is a.s. bounded, the norm of H(ω) is a.s. bounded by a constant, so its spectrum is contained in some nonrandom bounded interval I. Given an interval I⊂R, denote by C1(I) the set of all continuous functions ϕ on R with suppϕ⊂I and ϕ∞≤1.Theorem 28.
Consider the HamiltonianH(ω) of form (1) with a bounded random potential satisfying assumptions (W1)–(W3) and the interaction potential satisfying one of the assumptions (U0) and (U1). There is g0<+∞ such that if |g|≥g0, then the following property holds true.
For any bounded intervalI⊂R containing the a.s. spectrum of H(ω) and for all x,y∈Z>N, x≠y, and with the same c,a>0 as in (98),(99)Esupϕ∈C1I1xϕHω1y≤Ce-aln1+cρx,y.Proof.
We prove the claim under assumption (U0), using Corollary (97) established under this assumption. Similarly, case (U1) will follow from the analogous results proved in Section 7 for the infinite-range interactions.
Since bothV are U are bounded, the norm of the random operator H(ω) is a.s. bounded by a nonrandom constant; hence there exists a nonrandom bounded interval I⊂R containing a.s. the spectrum Σ(H(ω)). For any ball B and any points x,y∈B introduce a spectral measure μB,ωx,y uniquely defined by (100)∫ϕλdμB,ωx,yλ=1xϕHBωΠIHBω1y,where ϕ is an arbitrary bounded continuous function with suppϕ⊂I and similar spectral measures μωx,y for the operator H(ω) on the entire graph Z>N. If {BLk} is a growing sequence of balls, then μBLk,ωx,y converge vaguely to μωx,y as k→∞. Note that this fact holds true in a much more general context of unbounded operators: by a well-known result (cf. [24]), for the strong resolvent convergence of operators Hn→H, with a common core D, it suffices that Hnϕ→Hϕ for any element ϕ∈D; in turn, this implies the vague convergence of the spectral measures 〈φ,ϕ(Hn)ψ〉. In our case, we can choose as a core the subspace of all compactly supported functions, and on such functions, the operators Hn converge by stabilization.
So, by Fatou’s lemma on convergent measures, for any measurable setE⊂R and any growing sequence of balls BLk,(101)Eμωx,yE≤liminfk→∞EμBLkx,yE.Therefore, the uniform bounds in finite volumes BLk, established in Corollary 27, remain valid in the entire infinite graph.Now assertion (B) of the main Theorem 2 for finite-range interactions follows by taking continuous functions of the form (102)ϕt:λ⟼fIλeitλ,t∈R,where fI:R→[0,1] is an arbitrary continuous function interpolating between 1 on an interval I=[-R,R] containing a.s. the spectrum of H(ω) and 0 on R∖[-R-1,R+1].Note also that, arguing as in [23, Sect. 2.5], one can extend bound (99) to the compactly supported bounded Borel functions ϕ.


## 7. Adaptation to Infinite-Range Interactions

In this section, we need to adapt a certain number of definitions and statements to rapidly decaying, infinite-range interactions satisfying assumption (U1). In particular,(i)
given a numberδ∈(0,1/14), figuring in (U1), we set now(103)ϱ=2δ,α=1+4δ,τ=δ2and observe that α2<2, ϱ-δ>τ;(ii)
we replace the decay exponentγ(m,L)=m(1+L-τ), figuring in Definition 9 (cf. (48)), by (104)γm,L,n≔m1+L-τN-n+1.Definition 29.
A pair of ballsBL(x), BL(y) is called distant iff ρ(x,y)≥CNL1+δ, with CN=4N.Definition 30.
AnN-particle ball BL(x) is called partially interactive (PI) if diamu>2L1+δ and fully interactive (FI) otherwise.Lemma 31.
(A) If BL(x) is PI, then there exists a partition of [[1,N]] into nonempty complementary subsets J,Jc such that ρ(ΠJBL(x),ΠJcBL(x))>L1+δ. Consequently, BL(N)(x) admits a factorization (105)BLNx=BLn′x′×BLn′′x′′with dist(x′,x′′)>2L1+δ.
(B) If BL(x), BL(y) are two FI balls with ρ(x,y)>11NL, then ρ(ΠBL(x),ΠBL(y))≥L1+δ.The proof repeats verbatim that of Lemma20 in Section A.2, but with L replaced by L1+δ.The most important technical modification is required for Lemma21, the proof of which was very elementary, due to the use of eigenfunctions.Next, introduce the following finite-range approximationsUR, R≥r0, of a given interaction U generated by 2-body interaction potentials U(2)(r): (106)UR2r=1r≤RU2r. Then for any configuration x=(x′,x′′)∈Z>n′×Z>n′′ with ρ(x′,x′′)>R, the truncated interaction decomposes as follows: UR(x)=UR(x′)+UR(x′′). As a result, for any factorized subset of the form B=BL′(x′)×BL′′(x′′) with ρ(BL′(x′),BL′′(x′′))>R, the truncated Hamiltonian HB(R) admits the algebraic decomposition (107)HBR=HBL′x′R⊗1n′′+1n′⊗HBL′′x′′R.Next, we fix an arbitrarym≥1 and modify the definition of a PI ball:BL(N)(x) is called PI iff diam(x)>CN=4mN/m~.Therefore, a PI ball admits a factorizationBLN(x)=BL′(x′)×BL′′(x′′) with dist(x′,x′′)>CN/2NL=2mL/m~, so that, with Rk=CN/2NLk, one has (108)ϵRk<e-2mLk<12e-Lkβ.Lemma 32.
Assume that the interactionU satisfies condition (U1). Fix an energy E. Let an N-particle ball BLk(N)(u)=BLk(n′)(uJ)×BLk(n′′)(uJc) and let a sample of the random potential V(·;ω) be such that (a)
dist(x′,x′′)>Rk≔CN/2NLk;(b)
BLkn′(u′) and BLkn′′(u′′) are m-Loc;(c)
BLk(N)(u) is E-NR.
Then the ballBLk(N)(u) is (E,m)-NS.Proof.
SetRk=CN/2NLk and define U(Rk) and H(Rk) as above. Denote (109)Hu,kRk,N=HBLkNuRk,GE=HBLkNu-E-1,and G(Rk)(E)=(Hu,k(Rk,N)-E)-1. The operator Hu,k(Rk,N) admits the decomposition (110)Hu,kRk,N=HBLkn′u′⊗1n′′+1n′⊗HBLkn′x′′,with u=(u′,u′′)∈Zdn′×Zdn′′; thus its eigenvalues are the sums Ea,b=λa+μb, where {λa}=Σ(HBLk(u′)) is the spectrum of HBLk(u′) and, respectively, {μb}=Σ(HBLk(u′′)). Eigenvectors of Hu,k(Rk,N) can be chosen in the form Ψa,b=ϕa⊗ψb, where {ϕa} are eigenvectors of HBLkn′(u′) and {ψb} are eigenvectors of HBLkn′′(u′′). Note that, for each eigenvalue Ea,b=λa+μb, that is, for each pair (λa,μb), the nonresonance assumption |E-(λa+μb)|≥eLkβ reads as |(E-λa)-μb)|≥eLkβ and also as |(E-μb)-λa)|≥eLkβ. Therefore, we can write(111)GRku,y;E=∑λa∑μbϕau′ϕay′ψbu′′ψby′′λa+μb-E=∑λaPa′u′,y′GBLku′′Rku′′,y′′;E-λa=∑μbPb′′u′′,y′′GBLku′Rku′,y′;E-μb, where the resolvents GBLk(u′)(Rk)(E-μb) and GBLk(u′′)(Rk)(E-λa) are nonresonant: (112)GBLku′RkE-μb≤eLkβ,GBLku′′RkE-λa≤eLkβ. For any y∈∂-BLk(u), either ρ(u′,y′)=Lk, in which case we infer from (b)(113)GRku,y;E≤BLku′′e-γm,Lk,N-1Lk+Lkβ, or ρ(u′′,y′′)=Lk, and then we have (114)GRku,y;E≤BLku′e-γm,Lk,N-1Lk+Lkβ. In either case, the LHS is bounded by (115)exp-m1+Lk-τN-N-1+1Lk+Lkβ+ClnLk<12e-γm,Lk,N. Next, using the second resolvent identity, (116)G=GRk-GRkU-URkG, and the assumed E-NR property of the resolvents G, G(Rk), we conclude that (117)G-GRk≤U-URkGRkG≤e-2mLke2Lkβ≤12e-γm,Lk,N. Now the claim follows from the inequality (118)Gx,y;E-GRkx,y;E≤12e-γm,Lk,N,∀x,y∈BLk+1Nu.Lemma 33.
LetBLk+1(x),BLk+1(y) be two distant PI balls. Then, for L0 large enough,(119)PBLk+1x,BLk+1yareE,m-S<14Lk+1-PN,k+1.Proof.
Let(120)R2=∃E:BLk+1x,BLk+1yareE-R,S2=BLk+1x,BLk+1yareE,m-S.Then PS(2)≤PR(2)+PS(2)∖R(2), and within the event S(2)∖R(2) one of the balls BLk+1(x), BLk+1(y) must be E-NR, no matter how E∈R is chosen.
Next, consider the canonical factorization of the PI ball(121)BLk+1Nx=BLk+1n′x′×BLk+1n′x′′.By Lemma 32, if both BLk+1(x′) and BLk+1(x′′) are m-Loc and BLk+1(x) is (E,m)-S, then BLk+1(x) is E-R; hence within the complement of the event R(2), the other ball BLk+1(y) must be E-NR. On the other hand, consider the canonical factorization of the PI ball (122)BLk+1Ny=BLk+1n′y′×BLk+1n′′y′′.If both B(n′)Lk+1(y′) and B(n′′)Lk+1(y′′) are m-Loc and BLk+1(N)(y) is (E,m)-S, then BLk+1(N)(y) must be E-R. Let (123)L4=oneoftheballsBLk+1n′x′,BLk+1n′′x′′,BLk+1n′y′,BLk+1n′y′′ism-NLoc.Since these four balls correspond to systems with less than N particles, we can use induction in N and write (124)PL4≤4Lk+1-PN-1,k+1=4Lk+1-2PN,k+1.Then, as we have noticed, S(2)∖L(4)⊂R(2); hence, for L0 large enough, (125)PS2≤PR2+PL4≤e-Lkβ+4Lk+1-2PN,k+1<14Lk+1-PN,k+1.The statement of Lemma15 remains unchanged, but its proof requires a minor modification.Proof of Lemma15 under hypothesis (U1).
(A) By assumption, eitherBLk+1(u) is E-CNR and contains no (E,m)-S ball of radius Lk or there is a point w∈BLk+1(u) such that any ball BLk(v)⊂BLk+1(u) with ρ(w,v)≥CNLk1+δ is (E,m)-NS. In the former case, such an exclusion is unnecessary, but in order to treat both situations with one argument, we can formally set w=u (or any other point).
Fix pointsx,y with R≔ρ(x,y)>Lk1+ϱ, where ϱ=2δ is defined in (103). Note that for L0 large (hence, Lk also large), Lk1+ϱ≫Lk1+δ. By triangle inequality, (126)ρx,BCN-1Lkw+ρy,BCN-1Lkw≥R-2CN-2Lk1+δ.Assume first that (127)r′≔ρx,BCN-1Lk1+δw≥Lk+1,r′′≔ρy,BCN-1Lk1+δw≥Lk+1.All the balls of radius Lk both in Br′(x) and in Br′′(y) are automatically (E,m)-NS. Furthermore, r′+r′′≥R-2(CN-1)Lk1+δ-2≥R-2CNLk1+δ.
Consider the setB=Br′(x)×Br′′(y) and the function f:B→C defined by f(x′,x′′)=GBLk+1(x′,x′′;E). Since E is not a pole of the resolvent GBLk+1(u)(·), it is well defined (hence, bounded, on a finite set). By Lemma 10, f is (Lk,q)-dominated both in x′ and in x′′, with q≤e-γ(m,Lk,n). Therefore, one can write, with the convention -ln0=+∞, using Lemma 7 and setting for brevity J≔M-n+1,(128)-lnfu,y≥-lne-m1+1/2Lk-τJLkR-2CNLk1+δ-2Lk/Lk+1eLk+1β=m1+12Lk-τJLkLk+1R1-3CNRLk-Lk+1β=mR1+12Lk-τJ1-Lk-11-3CNLkLk1+ϱ-Lk+1βmR=mR1+12Lk-τJ1-Lk-11-3CNLk-ϱ-Lk+1βmLk1+ϱ≥m1+14Lk-τJR≥γm,Lk+1R.If r′=0 (resp., r′′=0), the required bound follows from the dominated decay of the function f(x′,x′′) in x′′ (resp., in x′).
(B) This assertion is proved in the same way as its counterpart in Section3.3.The rest of the scaling procedure presented in Section4 requires no modification and applies to infinite-range interactions.


## 8. Fermionic Hamiltonians on More General Graphs

The reduction to a standard lattice Laplacian on a subset{(x1,…,xN):x1<⋯<xN} with Dirichlet boundary conditions is no longer possible for particle systems on lattices Zd with d>1. Instead, one has to work with a symmetric power of the lattice, considered as a graph. So it seems reasonable to consider a fairly general, countable connected graph Z satisfying the condition of polynomial growth of balls:(129)∀x∈Z∀L≥1BLx≤CdLd,where BL(x)={y∈Z:dZ(x,y)≤L}. In particular, this gives a uniform bound on the coordination numbers, nZ(x)≤Cd (of course, this bound may be nonoptimal).An orthonormal basis in the Hilbert space of square-summable antisymmetric functionsΨ:ZN→C is formed by the functions (130)Φa=1N!∑π∈SN⨂j=1N1aπ-1j, where a={a1,…,aN} with #{a1,…,aN}=N.Further, define the graph(Z>N,E>N) as follows: the vertex set is (131)Z>N=a=a1,…,aN:aj∈Z,#a1,…,aN=N. Two vertices a, b form an edge iff (i)
the symmetric differencea⊖b has cardinality 1; that is, a={a1,c2,…,cN}, b={b1,c2,…,cN}, with #{a1,b1,c2,…,cN}=N+1;(ii)
dZ(a1,b1)=1.Next, define onZ>N the max-distance, (132)ρx,y=minπ∈SNmax1≤j≤NdZxπ-1j,yj, and introduce the balls BL(x) relative to the distance ρ(·,·).Now one can define the fermionic (negative) Laplacian(-Δ) on Z>N and random Hamiltonians H(ω)=-Δ+gV(x;ω)+U(x), where (133)Ux=∑i≠jUdxi,xj,U:N⟶R, and the external random potential energy, (134)Vx;ω=Vx1;ω+⋯+VxN;ω, is generated by a random field V:Z×Ω→R.The method presented in Sections3–7 applies to strongly disordered random Hamiltonians H(ω)=-Δ+gV(x;ω)+U(x) describing fermionic systems on connected graphs Z with polynomial growth of balls; indeed, one can see that we did not use particular properties of the one-dimensional lattice Z=Z other than polynomially bounded growth of balls.Strongly disordered bosonic systems can be treated in a similar way; the only modification required here concerns the explicit form of the matrix elements of the Laplacian, which remains a second-order finite-difference operator.



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*Source: 2129682-2016-02-18.xml*