Statistics of random quasi 1D Hamiltonian with slowly varying parameters. Painlev\'{e} again.

TL;DR

This paper analyzes the spectral statistics of random band matrices with slowly varying parameters, linking the Dyson equation near the spectrum edge to the Painlevé I equation, revealing connections to 2D gravity models.

Contribution

It demonstrates that the Dyson equation for such matrices reduces to the Painlevé I equation, establishing a novel connection between spectral statistics and 2D gravity.

Findings

Green function solution fixed by analytical properties

Painlevé I describes the spectrum edge behavior

Connection to random-matrix models of 2D gravity

Abstract

The statistics of random band--matrices with width and strength of the band slowly varying along the diagonal is considered. The Dyson equation for the averaged Green function close to the edge of spectrum is reduced to the Painlev\'{e} I equation. The analytical properties of the Green function allow to fix the solution of this equation. The former appears to be the same as that arose within the random--matrix regularization of 2d-gravity.