Asymptotic behaviors of the integrated density of states for random Schr odinger operators associated with Gibbs Point Processes

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Asymptotic behaviors of the integrated density of

states for random Schr¨odinger operators

associated with Gibbs Point Processes

Yuta Nakagawa∗†

October 21, 2022

Abstract

The asymptotic behaviors of the integrated density of states N(λ) of

Schr¨odinger operators with nonpositive potentials associated with Gibbs

point processes are studied. It is shown that for some Gibbs point pro-

cesses, the leading terms of N(λ) as λ↓ −∞ coincide with that for a Pois-

son point process, which is known. Moreover, for some Gibbs point pro-

cesses corresponding to pairwise interactions, the leading terms of N(λ)

as λ↓ −∞ are determined, which are diﬀerent from that for a Poisson

point process.

Keywords: Random Schr¨odinger operator; Density of states; Gibbs

point process

MSC2020 subject classiﬁcations: Primary 82B44, Secondary 60G55;

60G60; 60K35.

1 Introduction

We consider the random Schr¨odinger operator on L2(Rd, dx) deﬁned by

Hηω=−∆ + Vηω, Vηω(x) = ZRd

u0(x−y)ηω(dy),(1.1)

where u0is a nonpositive measurable function on Rd, and ηωis a point process

(see Section 2). We call u0the single site potential. We assume that ηωis

stationary and ergodic (see Section 2). The integrated density of states (IDS)

N(λ) of the Schr¨odinger operator is the nondecreasing function formally given

lim

L→∞

|ΛL|#{eigenvalues of HD

ηω,L less than or equal to λ},

∗Graduate School of Human and Environmental Studies, Kyoto University, Japan.

†E-mail: nakagawa.yuta.58n@st.kyoto-u.ac.jp

arXiv:2210.11381v1 [math.PR] 20 Oct 2022

where ΛLis the box (−L/2, L/2)d⊂Rd,|ΛL|denotes Lebesgue measure of

ΛL, and HD

ηω,L is the operator Hηωrestricted to ΛLwith Dirichlet boundary

condition. Because of the ergodicity of ηω, both N(λ) and the spectrum of

Hηωare independent of ωalmost surely. Moreover, N(λ) increases only on the

spectrum. See [2, 12] for the precise deﬁnition and the properties of the IDS.

The asymptotic behaviors of the IDS near the inﬁmums of the spectra are

well-studied. For a stationary Poisson point process (see [3] for the deﬁnition)

and a nonpositive and integrable single site potential u0which is continuous at

the origin and has a minimum value of less than zero there, the spectrum is R

apart from some exceptional cases (see [1] for details), and it holds that

log N(λ)∼ −λlog |λ|

u0(0) (λ↓ −∞),(1.2)

which is proved by Pastur in [11] (see also [12, (9.4) Theorem]), where we write

f(λ)∼g(λ) (λ↓ −∞) if f(λ)/g(λ) converges to one as λ↓ −∞. See [9] for

the asymptotic behavior for a singular nonpositive single site potential, and

[6, 10, 11] for that for a nonnegative single site potential.

The aim of this work is to investigate the asymptotic behaviors of N(λ)

as λ↓ −∞ for nonpositive single site potentials and Gibbs point processes:

point processes with interactions between the points. We mainly deal with

pairwise interaction processes (i.e. Gibbs point processes corresponding to pair-

wise interactions: the energy of the points {xj}is Pi<j ϕ(xi−xj), where ϕ

is a nonnegative symmetric function on Rdwith compact support). In [16],

Sznitman proved a result that is equivalent to determining the leading term of

the asymptotic behavior of the IDS for a nonnegative single site potential with

compact support and a pairwise interaction process.

From the proof of (1.2) in [12, (9.4) Theorem], we can ﬁnd that for a Poisson

point process, the probability that many points exist in a small domain aﬀects

the leading term of the IDS. Therefore, for a Gibbs point process correspond-

ing to a weak interaction that does not prevent the points from gathering (e.g.

Example 3.2), we expect that the IDS satisﬁes (1.2). This is proved in Theo-

rem 3.3. However, for a general Gibbs point process, (1.2) does not hold. In

Corollary 4.2, for a pairwise interaction process satisfying some conditions, we

determine the leading term of the asymptotic behavior of the IDS:

log N(λ)∼ − ϕ(0)

2ku0k2

λ2(λ↓ −∞),(1.3)

where ku0k2

S, deﬁned in Section 4, depends only on u0and the support Sof

ϕ. This implies that the IDS decays much faster than that for a Poisson point

process. In this case, because of the repulsion of the points, the probability that

some clusters of many points occur aﬀects the leading term of the IDS (see (4.5)

and (4.6)). The main tools for the proof of (1.3) are Proposition 3.4, which is

used in the proof of (1.2) in [12, (9.4) Theorem], and the upper estimate of the

Laplace functional in Proposition 4.6.

This paper is organized as follows. Section 2 introduces the Gibbs point

processes (see e.g. [3, 13, 14]). In Section 3, we prove that the leading term

of the asymptotic behavior of the IDS for a Gibbs point process corresponding

to a weak interaction is identical to (1.2). In Section 4, we treat a pairwise

interaction process satisfying some conditions and determine the leading term

of the asymptotic behavior of the corresponding IDS, which is diﬀerent from

(1.2).

2 Gibbs point processes

Let (C,F) denote the space of all locally ﬁnite measures on (Rd,B(Rd)) which

can be written as a countable sum Pn

j=1 δxj(n∈Z≥0∪ {+∞}, xj∈Rd, xi6=

xjfor any i6=j), equipped with the σ-algebra Fgenerated by {MΛ}Λ∈B(Rd),

where δxdenotes the Dirac measure at x∈Rd, and for every Λ ∈ B(Rd), MΛis

the function on Cdeﬁned by MΛ(η) = #(Λ ∩supp η). We note that all η∈ C is

simple (i.e. η({x})≤1 for any x∈Rd). We call a C-valued random element a

point process.

Let Cfbe the set of all ﬁnite measures in C. We introduce an energy function

to deﬁne the interaction between the points.

Deﬁnition 2.1. An energy function is a measurable function Ufrom Cfto

R∪ {+∞} such that:

•Umaps the null measure to 0;

•if U(η)=+∞, then U(δx+η) = +∞for all x∈Rd\supp η;

•U(τxη) = U(η) for any η∈ Cfand any x∈Rd, where τxη(·) = η(τ−x·),

and τxis the translation by vector x.

We introduce a pairwise energy function: an energy function corresponding

to a pairwise interaction.

Deﬁnition 2.2. An energy function deﬁned by

U(η) = X

{x,y}⊂ supp η

x6=y

ϕ(x−y) (2.1)

is called a pairwise energy function, where ϕis a measurable and symmetric

(i.e. ϕ(x) = ϕ(−x)) function from Rdto R≥0∪ {+∞} with compact support.

For a pairwise energy function Udeﬁned by (2.1) and a bounded Λ ∈ B(Rd),

we deﬁne

UΛ(η) = X

{x,y}⊂ supp η

{x,y}∩Λ6=∅, x6=y

ϕ(x−y) (η∈ Cf).

For an energy function Uexcept for pairwise energy functions and a bounded

Λ∈ B(Rd), we set

UΛ(η) = U(η)−U(ηΛc) (η∈ Cf),

where ηΛdenotes the measure η(· ∩ Λ), Λc=Rd\Λ, and ∞−∞ = 0 for

convenience. We can see UΛ(η) means the variation of the energy when adding

ηΛto ηΛc.

In this paper, we assume the ﬁnite range property:

Deﬁnition 2.3. We say that an energy function Uhas a ﬁnite range R > 0 if

for all bounded Λ ∈ B(Rd) and all η∈ Cf, it holds that

UΛ(η) = UΛ(ηΛ+B(0,R)),

where B(x, R) is the closed ball centered at xwith radius R, and Λ + B(0, R)

denotes the set {x+y|x∈Λ, y ∈B(0, R)}.

A pairwise energy function deﬁned by (2.1) has a ﬁnite range R > 0 if and

only if the support of ϕis included in B(0, R).

For an energy function Uwith a ﬁnite range R > 0, we can deﬁne

UΛ(η) = UΛ(ηΛ+B(0,R)), h(x, η) = U{x}(η+δx) (η∈ C, x ∈Rd\supp η).

The function his called the local energy function, which means the variation of

the energy when adding δxto η.

Let FΛdenote the σ-algebra generated by {MΛ0}Λ0⊂Λ,Λ0∈B(Rd). When an

energy function Uhas a local energy function bounded below, for every γ∈ C

and every bounded Λ ∈ B(Rd) with positive Lebesgue measure, we deﬁne the

probability measure PΛ,γ on (C,FΛ) by

PΛ,γ (dη) = 1

ZΛ(γ)e−UΛ(ηΛ+γΛc)π1(dη),

where πzis the distribution of the Poisson point process with intensity z(see

[3] for the deﬁnition), and ZΛ(γ) is a normalizing constant:

ZΛ(γ) = Ze−UΛ(ηΛ+γΛc)π1(dη).

We see

ZΛ(γ)≥Z{MΛ=0}

π1(dη) = e−|Λ|>0.(2.2)

Moreover, if the local energy function his bounded below by a∈R, since for

any distinct points x1, . . . , xn∈Λ,

UΛ(

j=1

δxj+γΛc) =

j=1

h(xj,

j−1

i=1

δxi+γΛc)≥an,

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AsymptoticbehaviorsoftheintegrateddensityofstatesforrandomSchrodingeroperatorsassociatedwithGibbsPointProcessesYutaNakagawa*October21,2022AbstractTheasymptoticbehaviorsoftheintegrateddensityofstatesN()ofSchrodingeroperatorswithnonpositivepotentialsassociatedwithGibbspointprocessesarestudied.Itis...

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Asymptotic behaviors of the integrated density of states for random Schr odinger operators associated with Gibbs Point Processes

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