Statistical Properties of Random Banded Matrices with Strongly Fluctuating Diagonal Elements

TL;DR

This paper analyzes the statistical properties of a special class of random banded matrices with highly fluctuating diagonal elements, confirming recent findings about their local density of states and eigenfunction structure through analytical methods.

Contribution

It introduces an analytical approach using a supersymmetric nonlinear sigma-model to study RBMs with strongly fluctuating diagonals, extending previous models.

Findings

Confirmed Lorentzian form of local density of states

Validated two-scale spatial structure of eigenfunctions

Provided analytical insights into eigenfunction component distributions

Abstract

The random banded matrices (RBM) whose diagonal elements fluctuate much stronger than the off-diagonal ones were introduced recently by Shepelyansky as a convenient model for coherent propagation of two interacting particles in a random potential. We treat the problem analytically by using the mapping onto the same supersymmetric nonlinear $\sigma-$model that appeared earlier in consideration of the standard RBM ensemble, but with renormalized parameters. A Lorentzian form of the local density of states and a two-scale spatial structure of the eigenfunctions revealed recently by Jacquod and Shepelyansky are confirmed by direct calculation of the distribution of eigenfunction components.