Delocalization for random displacement models with Dirac masses

TL;DR

This paper investigates how weak disorder affects eigenfunction localization in a 2D random Schrödinger operator with Dirac delta potentials on a large torus, revealing a transition from exponential to polynomial decay above a certain energy.

Contribution

It establishes the breakdown of exponential localization at high energies for a 2D random delta potential model with weak disorder, linking eigenfunction decay to lattice point problems.

Findings

Exponential localization breaks down above energy E_0.

Eigenfunctions exhibit polynomial decay at high energies.

Results connect eigenfunction distribution to lattice point problems.

Abstract

We study a random Schroedinger operator, the Laplacian with random Dirac delta potentials on a torus T^d_L = R^d/LZ^d, in the thermodynamic limit L\to\infty, for dimension d=2. The potentials are located on a randomly distorted lattice Z^2+\omega, where the displacements are i.i.d. random variables sampled from a compactly supported probability density. We prove that, if the disorder is sufficiently weak, there exists a certain energy threshold E_0>0 above which exponential localization of the eigenfunctions must break down. In fact we can rule out any decay faster than a certain polynomial one. Our results are obtained by translating the problem of the distribution of eigenfunctions of the random Schroedinger operator into a study of the spatial distribution of two point correlation densities of certain random superpositions of Green's functions and its relation with a lattice point problem.