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

TL;DR

This paper introduces a new probabilistic expression for the eigenvalues of 1D Anderson operators using two boundary-starting processes, revealing the exponential decay behavior of eigenvector tails in finite intervals.

Contribution

It provides a novel formula linking eigenvalues to forward-backward random processes, advancing understanding of spectral properties in the Anderson model.

Findings

Eigenvalue distribution expressed via boundary processes

Eigenvector tails decay approximately exponentially with Brownian motion correction

Results extend to critical scaling case with potential multiplied by 1/√N

Abstract

We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval $[0,N]$, 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 approximatelylike $\exp(\sigma B\_{|n-k|}-\gamma\frac{|n-k|}{4})$ where $B\_{s}$ is the Brownian motion and $k$ is uniformly chosen in $[0,N]$ independentlyof $B\_{s}$. 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 $\frac{1}{\sqrt{N}}$