Spectral Statistics for one dimensional Anderson model with unbounded but decaying potential

TL;DR

This paper investigates the spectral statistics of a one-dimensional Anderson model with decaying, unbounded potential, showing that under certain conditions, the eigenvalue process behaves like a clock process.

Contribution

It demonstrates that for a specific decay rate and fat-tailed distribution, the eigenvalue process in the Anderson model converges to a clock process, extending understanding of spectral statistics with unbounded potentials.

Findings

Eigenvalue process in (-2,2) is a clock process under certain conditions.

Spectral statistics are analyzed for unbounded, decaying randomness.

Conditions on decay rate and tail behavior are crucial for results.

Abstract

In this work, we study the spectral statistics for Anderson model on $\ell^2(\mathbb{N})$ with decaying randomness whose single site distribution has unbounded support. Here we consider the operator $H^\omega$ given by $(H^\omega u)_n=u_{n+1}+u_{n-1}+a_n\omega_n u_n$, $a_n\sim n^{-\alpha}$ and $\{\omega_n\}$ are real i.i.d random variables following symmetric distribution $\mu$ with fat tail, i.e $\mu((-R,R)^c)<\frac{C}{R^\delta}$ for $R\gg 1$, for some constant $C$. In case of $\alpha-\frac{1}{\delta}>\frac{1}{2}$, we are able to show that the eigenvalue process in $(-2,2)$ is the clock process.