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 $[0,N]$, 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 $\frac{1}{\sqrt{N}}$