Distribution of Localization Centers in Some Discrete Random Systems

Abstract

As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l 2 (Z d ) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible. Mathematics Subject Classification (2000): 82B44, 81Q10where ν is the distribution of V ω (0) [15].(2)(Anderson localization) There is an open interval I ⊂ Σ such that with probability one, the spectrum of H ω on I is pure point with exponentially decaying eigenfunctions. I can be taken (i) I = Σ if λ is large enough, (ii) on the band edges, and (iii) away from the spectrum of the free Laplacian if λ sufficiently small (e.g., [7,20,1,2]).Recently, some relations between the eigenvalues and the corresponding localization centers are discussed [17]. It roughly implies,Hence the distribution of the localization centers are "thin" in space 1 .(Hence the localization centers are repulsive if the energies get closer.On the other hand, in [10], they study the "natural scaling limit" of the random measure in R d+1 (the product of energy and space) composed of the eigenvalues and eigenfunctions. The result there roughly implies that their distribution with eigenvalues in the order of L −d from the reference energy E 0 , and with eigenfunctions in the order of L from the origin, obey the Poisson law on R d+1 . This work can also be regarded as an extension of the work by Minami [16] who showed that the point process on R composed of the eigenvalues of H in the finite volume approximation converges to the Poisson process on R. To summarize, [17,10] imply that the eigenfunctions whose energies are in the order of L −d are non-repulsive while those in the order of L −2d are repulsive, which are consistent with Minami's result [16].The aim of this paper is to supplement [10] from a technical point of view : (i) to study the distribution of the localization centers which is technically different from what is done in [10], and (ii) to study what can be said for those models in which Minami's estimate and the fractional moment bound, which are the main tool in [10], are currently not known to hold.We set some notations. Notation :1 This result follows easily from the upper bound on the density of states. So in the Lifschitz tail region, we have |x(E)| ≥ (const.)e (const.)L d 2 2if |E − E 0 | ≤ L

The scaling limit of eigenfunctions for 1d random Schrödinger operator

“…When α < 1/2, this result is the same as that in [2,5], while for α > 1/2 this result is natural. For α = 1/2, this result implies that, the localization center U of the eigenfunction ψ is uniformly distributed and ψ has the power law decay around U with Brownian fluctuation.…”

Global multiplicity bounds and spectral statistics for random operators

“…The first rigorous work of Molchanov [33] led later to the Minami Theory [34], which establishes a set of sufficient conditions for the local spectral statistics to be Poisson. There are several papers on local spectral statistics on discrete models such as [4,35,36,37,38,39,40,41,42,43,44,45]. Dietlein and Elgart showed Minami like estimate and thereby showing Poisson local statistics at the spectral edge in case of random Schrödinger operators in [46].…”

The Parabolic Anderson Model

“…The main value of [Min96] is the introduction of a flexible estimate that establishes the existence of a gap between two subsequent eigenvalues, an estimate that is now called the Minami estimate. The first result on the convergence of point processes of both the eigenvalues and the concentration centres of the eigenfunctions is [KilNak07]; see also [Nak07]. The currently strongest available results are in [GerKlo14] and [GerKlo13], where [GerKlo14] works in the bulk of the spectrum and [GerKlo13] close to the top; see also [GerKlo11].…”