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

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.

arXiv comment16 pages