Summing graphs for random band matrices

TL;DR

This paper develops a resummation method for perturbation series in random band matrices, analyzing spectral properties, localization, and correlations, with extensions to lattice Hamiltonians with long-range hopping.

Contribution

It introduces a diagrammatic resummation approach using topological classification for random band matrices, providing new insights into spectral edge behavior and eigenvector localization.

Findings

Derived asymptotics for energy level correlations.

Estimated localization length of eigenvectors.

Extended analysis to d-dimensional lattice Hamiltonians.

Abstract

A method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices. The topological classification of Feynman diagrams, which was actively used in last years for matrix model regularization of 2d-gravity, turns out to be very useful for band matrices. The critical behavior at the edge of spectrum and the asymptotics of energy level correlation function are considered. This correlation function together with the hypothesis about universality of spectral correlations allows to estimate easily the localization length for eigen-vectors. A smoothed two-point correlation function of local density of states $\overline{ \rho(E_1,i) \rho(E_2,j)_c}$ as well as the energy level correlation for finite size band matrices are also found. As d-dimensional generalization of band matrices lattice Hamiltonians with long-range random hopping are considered as well.