A forward-backward random process for the spectrum of 1D Anderson operators

Abstract

We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval $[1,N]$, in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximately like $\exp (\sigma B_{|n-k|}-\gamma \frac{|n-k|} {4})$ where $\gamma , \sigma >0 $, $B_{s}$ is the Brownian motion and $k$ is uniformly chosen in $[1,N]$ independently of $B_{s}$. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random potential is multiplied by a factor $\frac{1} {\sqrt{N} }$.