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

The asymptotic behaviors of the integrated density of states $N(\lambda)$ of Schr\"odinger operators with nonpositive potentials associated with Gibbs point processes are studied. It is shown that for some Gibbs point processes, the leading terms of $N(\lambda)$ as $\lambda\downarrow-\infty$ coincide with that for a Poisson point process, which is known. Moreover, for some Gibbs point processes corresponding to pairwise interactions, the leading terms of $N(\lambda)$ as $\lambda\downarrow-\infty$ are determined, which are different from that for a Poisson point process.


Introduction
We consider the random Schrödinger operator on L 2 (R d , dx) defined by where u 0 is a nonpositive measurable function on R d , and η ω is a point process (see Section 2).We call u 0 the single site potential.We assume that η ω is stationary and ergodic (see Section 2).The integrated density of states (IDS) N (λ) of the Schrödinger operator is the nondecreasing function formally given by lim #{eigenvalues of H D ηω,L less than or equal to λ}, where Λ L is the box (−L/2, L/2) d ⊂ R d , |Λ L | denotes Lebesgue measure of Λ L , and H D ηω,L is the operator H ηω restricted to Λ L with Dirichlet boundary condition.Because of the ergodicity of η ω , both N (λ) and the spectrum of H ηω are independent of ω almost 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 u 0 which 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 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 pairwise interactions: the energy of the points {x j } is i<j ϕ(x i − x j ), where ϕ is a nonnegative symmetric function on R d with 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 corresponding 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 Theorem 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: where u 0 2 S , defined in Section 4, depends only on u 0 and the support S of 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).

Gibbs point processes
Let (C, F) denote the space of all locally finite measures on (R d , B(R d )) which can be written as a countable sum x j for any i = j), equipped with the σ-algebra F generated by {M Λ } Λ∈B(R d ) , where δ x denotes the Dirac measure at x ∈ R d , and for every Λ ∈ B(R d ), M Λ is the function on C defined by M Λ (η) = #(Λ ∩ supp η).We note that all η ∈ C is simple (i.e.η({x}) ≤ 1 for any x ∈ R d ).We call a C-valued random element a point process.
Let C f be 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 U from C f to R ∪ {+∞} such that: • U maps the null measure to 0; and τ x is 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 is called a pairwise energy function, where ϕ is a measurable and symmetric (i.e.ϕ(x) = ϕ(−x)) function from R d to R ≥0 ∪ {+∞} with compact support.
For a pairwise energy function U defined by (2.1) and a bounded Λ ∈ B(R d ), we define For an energy function U except for pairwise energy functions and a bounded Λ ∈ B(R d ), we set where η Λ denotes the measure η(• ∩ Λ), Λ c = R d \ Λ, and ∞ − ∞ = 0 for convenience.We can see U Λ (η) means the variation of the energy when adding η Λ to η Λ c .
In this paper, we assume the finite range property: Definition 2.3.We say that an energy function U has a finite range R > 0 if for all bounded Λ ∈ B(R d ) and all η ∈ C f , it holds that where B(x, R) is the closed ball centered at x with radius R, and Λ + B(0, R) A pairwise energy function defined by (2.1) has a finite range R > 0 if and only if the support of ϕ is included in B(0, R).
For an energy function U with a finite range R > 0, we can define The function h is called the local energy function, which means the variation of the energy when adding δ x to η.Let F Λ denote the σ-algebra generated by {M Λ } Λ ⊂Λ, Λ ∈B(R d ) .When an energy function U has a local energy function bounded below, for every γ ∈ C and every bounded Λ ∈ B(R d ) with positive Lebesgue measure, we define the probability measure P Λ,γ on (C, F Λ ) by where π z is the distribution of the Poisson point process with intensity z (see [3] for the definition), and Z Λ (γ) is a normalizing constant: We see Moreover, if the local energy function h is bounded below by a ∈ R, since for any distinct points x 1 , . . ., x n ∈ Λ, we have Now we introduce the Gibbs point processes.
Definition 2.4.We assume that an energy function U has a finite range, and its local energy function is bounded below.A point process is a Gibbs point process for the energy function U if its distribution P on C satisfies the following conditions: • for all bounded Λ ∈ B(R d ) with positive Lebesgue measure and all bounded F-measurable function f , it holds that (2.4) The equation (2.4) is called the Dobrushin-Lanford-Ruelle equation (DLR equation).We call a Gibbs point process for a pairwise energy function a pairwise interaction process.We note that if U is identically zero, the corresponding Gibbs point process is the Poisson point process with intensity one.
We say a point process is stationary if its distribution P satisfies P (τ x A) = P (A) for all x ∈ R d and all A ∈ F. Under the assumption in Definition 2.4, at least one stationary Gibbs point process for U exists (see [4,5]).
A point process with a distribution P is called ergodic if P (A) is either zero or one for any A ∈ F such that P (A τ x A) = 0 for all x ∈ R d , where A B denotes the symmetric difference of A and B. If an energy function with a finite range R > 0 has a local energy function bounded below by a ∈ R d , and R d e −a is sufficiently small, then the corresponding DLR equation has a unique solution (see [8] for more details).The uniqueness guarantees the stationarity and ergodicity of the corresponding Gibbs point process (see [13,Section 4]).
We end this section by introducing stochastic domination.We say that a measurable function f on (C, F) is increasing, if f (η) ≤ f (γ) whenever supp η ⊂ supp γ.If an energy function U has a local energy function bounded below by − log z for some z > 0, then the corresponding probability measure P Λ,γ is stochastically dominated by π z : for any bounded Λ ∈ B(R d ) and any bounded (see [7] for the proof).If there is a Gibbs point process for U , then for its distribution P , we also have from the DLR equation.

Weak interactions
As we see below, for a bounded integrable single site potential and a stationary and ergodic Gibbs point process with a local energy function bounded below, the Schrödinger operator (1.1) is essentially self-adjoint on C ∞ c almost surely, and the corresponding IDS exists, where C ∞ c denotes the set of all smooth functions on R d with compact support.We note that all Schrödinger operators in this paper have these properties.
Let P be the distribution of the Gibbs point process.From (2.5), for any p ∈ [1, ∞) and any bounded Λ ∈ B(R d ), we have where z > 0 is a constant such that the local energy function is bounded below by − log z.
where x denotes the least integer greater than or equal to x.
Condition (W) implies that the interaction between many points in a small region is sufficiently weak.Example 3.2 (Pairwise energy function).In general, the local energy functions of the pairwise energy functions are unbounded.However, a pairwise energy function such that the closer the two points approach, the weaker the interaction gets, could satisfy condition (W).For example, a pairwise energy function defined by (2.1) such that ϕ The main result in this section is the following.
Theorem 3.3.Let an energy function U with a finite range satisfy condition (W), its local energy function be bounded below, and a single site potential u 0 be nonpositive, integrable, continuous at the origin, and have a minimum value of less than zero there.We assume that a Gibbs point process for U is stationary and ergodic.Then the corresponding IDS satisfies that The asymptotic behavior (3.2) coincides with (1.2): the asymptotic behavior of the IDS for a Poisson point process.
One of the main tools in this paper is the following, which is proved by the path integral representation of the Laplace transform of the IDS and Dirichlet-Neumann bracketing (see [12, (9.1) Theorem and (9.2) Theorem] and [2] for the proof).
Proposition 3.4.For a stationary and ergodic Gibbs point process with a local energy function bounded below and a bounded integrable single site potential u 0 , the following hold: • for any ε > 0 and any δ ∈ (0, 1/2), there exists λ 0 < 0 such that for any λ < λ 0 , log N (λ) • for any t > 1 and any λ ∈ R, where P is the distribution of the Gibbs point process.
The integral on the right-hand side of (3.4) is called the Laplace functional.
Proof of Theorem 3.3.Let P be the distribution of the Gibbs point process.
(Lower bound) For simplicity, we put u(x) = −u 0 (−x) (i.e.u is nonnegative, and V η (x) = − u(y − x)η(dy)).We assume that the energy function U has a finite range R > 0, and the local energy function is bounded below by − log z for some z > 0.
Fix nonempty bounded Λ ∈ B(R d ) and n ∈ N.
For all η ∈ A and P -almost all γ ∈ C, we have Hence, from the DLR equation and (2.3), we obtain where 1 A is the indicator function of A.
From condition (W), we have Hence, we obtain lim inf where we use Stirling's formula in the last equation.This implies that lim inf .

Main results
In this section, we consider a pairwise interaction process corresponding to a pairwise interaction such that the repulsion does not fade as the two points approach each other (cf.Example 3.2).For such a pairwise interaction process, the leading term of the asymptotic behavior of the IDS could be affected by the values of the single site potential that are not necessarily the minimum value (cf.(1.2)).To represent the effect, we define for every bounded S ∈ B(R d ) and every measurable function u on R d .We introduce the following condition associated with the range of the interaction.
(S) S ∈ B(R d ) is nonempty, bounded and symmetric (i.e. S = −S), and for any α ∈ (0, 1), where Int S is the interior of S, and for a ∈ R, aS denotes the set {ax | x ∈ S}.
We remark that S ∈ B(R d ) satisfying condition (S) is a bounded star domain (i.e.αS ⊂ S for any α ∈ (0, 1)) including the origin in its interior; however, the converse is false.
The following theorem gives the upper and lower estimates of the asymptotic behavior of the IDS for a pairwise interaction process.Theorem 4.1.Let S ∈ B(R d ) satisfy condition (S), and u 0 be a continuous integrable nonpositive single site potential satisfying 0 < u 0 2 S < ∞.Moreover, we assume that a pairwise interaction process for an energy function U defined by (2.1) having a finite range is stationary and ergodic.Then, we have the following: (a) If ϕ is upper semi-continuous at the origin, ϕ(0) > 0, and ϕ(x) = 0 whenever x ∈ S c , then it holds that If u 0 has a compact support, and there exists a positive constant a such that ϕ(x) ≥ a whenever x ∈ S, then it holds that From Theorem 4.1, we determine the leading terms of the asymptotic behaviors of the IDS for some pairwise interaction processes: Corollary 4.2.Let a single site potential u 0 be a nonpositive continuous function that is not identically zero, with compact support, and ϕ be continuous at the origin and satisfy that ϕ(0) > 0, ϕ(x) ≥ ϕ(0) for any x ∈ S, and ϕ(x) = 0 for any x ∈ S c , where S ∈ B(R d ) satisfies condition (S).We assume that a pairwise interaction process for a pairwise energy function defined by (2.1) is stationary and ergodic.Then the corresponding IDS satisfies that Remark 4.3.When ϕ is not upper semi-continuous at the origin and is bounded on a neighborhood there, we can apply Theorem 4.1 (a) by replacing the value of ϕ(0) so that ϕ is upper semi-continuous at the origin.Similarly, Corollary 4.2 could be applied by replacing ϕ(0) so that ϕ is continuous at the origin.These are justified by the fact that the pairwise interaction processes are independent of the value of ϕ(0).for some a, R ∈ (0, ∞)) introduced in [15].In this remark, we fix a single site potential satisfying the condition of Corollary 4.2.Theorem 4.1 implies that for a pairwise interaction process such that c 1 < ϕ < c 2 on a neighborhood of the origin for some c 1 , c 2 ∈ (0, ∞), the corresponding IDS satisfies that For a hardcore process (i.e.ϕ = +∞ on a neighborhood of the origin), the corresponding IDS does not satisfy (4.3).In this case, there is a constant β such that V η > β for P -almost all η ∈ C, where P is the distribution of the hardcore process, and hence N (λ) = 0 whenever λ is less than β.

Proof of Theorem 4.1 (a)
To apply (3.3), we use (4.5) unlike either case of a Gibbs point process in Section 3 or a Poisson point process.Let P be the distribution of the pairwise interaction process in Theorem 4.1 (a).

Proof of Theorem 4.1 (b)
For the proof, we set the simple functions that approximate u, where u(x) = −u 0 (−x).For every n ∈ N, we put where u n,j = sup y∈Λ n,j u(y), and Λ n,j denotes the box {x+j/n | x ∈ (0, 1/n] d }.Proof.We note that u is a nonnegative continuous function with compact support.Fix a constant b > 0. It is easy to see that for all sufficiently large n ∈ N, Since dist(S c , αS) > 0 for any α ∈ (0, 1), we can find a monotonically nonincreasing sequence (β n ) ∞ n=n0 (n 0 ≥ 1) convergent to one, such that dist(S c , 1 We fix ε > 0. We find L > 0 such that supp u n ⊂ B(0, L) for all n ∈ N.For each n ≥ n 0 , we note that u n S b/n < ∞, and choose finite points We can find a constant K > 0 independent of ε such that k n ≤ K for all n ≥ n 0 .We put y n,j = β n x n,j .From (4.9), we get Since u is uniformly continuous, and u n is uniformly convergent to u, there exists δ > 0 such that for all sufficiently large n ∈ N, |u n (x) − u(y)| < ε whenever |x − y| < δ.Hence, from (4.10) and (4.11), for all sufficiently large n ∈ N, we obtain where m is the maximum value of u.The lemma is proved by combining this and (4.8).
Let P be the distribution of the Gibbs point process in Theorem 4.1 (b).The key to the proof of Theorem 4.1 (b) is the next proposition: the upper estimate of the Laplace functional as t → ∞.Proposition 4.6.We put where v 1 , . . ., v k > 0, Λ 1 , . . ., Λ k ∈ B(R d ) are disjoint bounded sets with positive Lebesgue measure such that for each j = 1, . . ., k, x − y ∈ S whenever x, y ∈ Λ j .Then we have where K denotes the set {J ⊂ {1, . . ., k} |for all i = j ∈ J, there are x ∈ Λ i and y ∈ Λ j such that x − y ∈ S c }.
The proof of Proposition 4.6 proceeds via two lemmas.
The idea of the proof is to change of variables: where we put Proof of Lemma 4.8.We fix ε > 0. Let G(t; I) denote the left-hand side of (4.13).We have for all sufficiently large t > 0. Now we investigate the case that I = ∅.We assume that (1, 2) ∈ I without loss of generality.We consider in the left-hand side of (4.13), where we put where I(J) = {(i, j) ∈ J × J | i < j, and x − y ∈ S for any x ∈ Λ i , y ∈ Λ j }.From Lemma 4.8, for any nonempty subset J ⊂ {1, . . ., k} and all sufficiently large t > 0, we obtain where K(J) = {J ⊂ J | (i, j) = I(J) whenever i, j ∈ J }. Since K(J) ⊂ K, we obtain (4.12).
Proof of Theorem 4.1 (b).We put u(x) = −u 0 (−x) and u n as (4.7).We note that for each n ∈ N, u n,j = 0 except for finitely many j ∈ Z d .We fix ε > 0.
(W) There exists a positive constant ε 0 and a positive function r on (1, ∞), such that lim x→∞ r(x) = 0, lim x→∞ log r(x) log x = 0, and for any 0

Example 3 . 1 (
Energy function with a bounded local energy function).Some energy functions have bounded local energy functions.An example is an area energy function: j , R) , which has the local energy function h satisfying 0 ≤ h ≤ |B(0, R)|, where R > 0 is a constant.If a local energy function h of an energy function U is bounded above by a ∈ R, then we have U (η) ≤ aM R d (η) for all η ∈ C f .Hence, the energy function U satisfies condition (W).

Lemma 4 . 5 .lim n→∞ u n 2 S
Let S ∈ B(R d ) satisfy condition (S).Then for any b > 0, it holds that b/n = u 2 S , where S b/n denotes the set (S c + B(0, b/n)) c .