Fluctuations and localization length for random band GOE matrix

Abstract

We prove that GOE random band matrix localization length is $\le C\left(\log W\right)^3 W^2$, where $W$ is the width of the band and $C$ is an absolute constant. Our method consists of Green function edge-to-edge vector action approach to the Schenker method. That allows to split and decouple the action, so that it becomes transparent that \emph{the magnitudes of two consecutive Schur complements vector actions can not be both larger than an absolute constant}. That is the central technological ingedient of the method. It comes from rather involved estimates $($ the main estimates of the metod $)$, in combination with an equation relating two magnitudes in question. We call the latter \emph{recurrence equation}. The method results in the \emph{lower bound of the variance of the $\log$--norm of the vector action at $\gtrsim NW^{-1}$}, where $N$ is the total number of GOE blocks, condition $N\lesssim W^D$ with an absolute constant $D\gg 1$ applies.