Traffic distributions of random band matrices

TL;DR

This paper explores the traffic probability framework for random band matrices, comparing them to Wigner matrices, and establishes new concentration inequalities and deviations from free probability universality.

Contribution

It extends traffic probability analysis to random band matrices, revealing their deviation from free probabilistic universality and establishing new concentration results.

Findings

Traffic space of Wigner matrices realizes the traffic CLT.

Random band matrices deviate from free probabilistic universality.

Established Markov-type concentration inequalities for traffic distributions.

Abstract

We study random band matrices within the framework of traffic probability, an operadic non-commutative probability theory introduced by Male based on graph operations. As a starting point, we revisit the familiar case of the permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We then show how the traffic space of Wigner matrices completely realizes the traffic central limit theorem. We further prove general Markov-type concentration inequalities for the joint traffic distribution of independent Wigner matrices. We then extend our analysis to random band matrices, as studied by Bogachev, Molchanov, and Pastur, and investigate the extent to which the joint traffic distribution of independent copies of these matrices deviates from the Wigner case.