We assess the probability of resonances between sufficiently distant states of an

We study quantum systems in a disordered environment, usually referred to as Anderson-type models, due to the seminal paper by Anderson [

The mathematical Anderson localization theory has motivated a large number of studies of random differential and finite-difference operators during the last forty years, but only recently a significant progress has been made in the rigorous theory of multiparticle quantum systems in a disordered environment with a nontrivial interparticle interaction (cf. [

Specifically, we consider a system of

Our main assumptions on the random field

We focus on the probabilistic eigenvalue concentration bounds, known as the Wegner-type bounds, due to the celebrated paper by Wegner [

The role and importance of such bounds can be easily understood: the MSA procedure starts with the analysis of the resolvents

We introduce the graph structure in the configuration space

In [

Starting from

In [

Despite many differences between the MSA and the FMM, similar technical difficulties have been encountered in both cycles of papers. Namely, it turned out to be difficult to prove the decay bounds of eigenfunctions

However, due to a highly correlated nature of the potential of a multiparticle system, even the above concession did not suffice, and it was easier to use the “Hausdorff distance” (which is a pseudometric) between the points

Aizenman and Warzel [

As a result, one could not prove complete localization, say, in a finite cube

In the present paper we address this problem and show that abnormally strong resonances between distant states in the configuration space, related by partial charge transfer processes, are unlikely and prove efficient probabilistic estimates for such unlikely situations.

In this paper, we work with connected, locally finite graphs

We assume that the growth rate of the balls in

We consider in particular the class

We assume that an IID random field

Introduce the following notations. Given a finite subset

We will assume that the random field

(RCM):

In fact, for the applications to the MSA, it suffices to require the bound (

In the particular case of a Gaussian IID field

In general, the conditional probability distribution function

Let

Formally speaking, the condition (RCM) does not refer to the growth rate of balls in the graph

Let

The following statement establishes the validity of the condition (RCM) for a class of IID random potentials, including the popular in physics uniform distribution in a bounded interval.

Consider an IID random field

Given a connected graph

Intervals of integer values will often appear in our formulae, and it is convenient to use the standard notation

We identify

Pictorially,

The graph structure defines in

One can also define the max-distance on

It turns out that the graph distance

On the other hand, this technical problem is merely an artefact of the language of

Let

By a slight abuse of notation, we will identify a subconfiguration

(a) Let

(b) The support

(c) Given an index subset

A ball

The physical meaning of the weak separation is that in a certain region of the one-particle configuration space the presence of particles from

In fact, (

Any pair of

Given

Let

Further, denote

Introduce the occupation numbers of the sets

There can be two possible situations.

For all

yielding, by (

If

For some

Since not all the summands

Setting

Let

Let

The operators

Let

By hypothesis, we have

The main EVC bound, established for the Hamiltonian

A technically convenient alternative to restricting

The subspace of all antisymmetric square-summable functions

Up to a constant factor of

Similarly, the subspace of symmetric functions

and the combinatorial weight

In a more general case, there are alternative constructions of the reduced graph.

The standard construction of a symmetric power of an arbitrary locally finite graph

An alternative construction, which we present first in the fermionic case, is easily adapted to the bosonic systems.

A configuration of

For example, with

The vertex set of the

Consider the integer-valued functions

For example,

The vertex set of the

The following formula gives an equivalent form of the restriction of the

We focus now on the fermionic case, which corresponds to the physical model of

The choice of the metric, defining the notion of a ball in the

The following example clearly explains the difference between the fermionic ball

Let

Let

The claim follows from Theorem

The author declares that there is no conflict of interests regarding the publication of this paper.

It is a pleasure to thank Werner Kirsch and the Fern Universität Hagen for their warm hospitality and simulating atmosphere at the PasturFest (May 2013) and Abel Klein, Fumihiko Nakano, Nariyuki Minami, Stanislav A. Molchanov, Günter Stolz, and Ivan Veselić for fruitful discussions.