A renormalisation group approach to the universality of Wigner's semicircle law for random matrices with dependent entries

TL;DR

This paper demonstrates that under certain scaling bounds on cumulants, Wigner's semicircle law applies to random matrices with dependent entries, using a renormalisation group approach and replica technique.

Contribution

It introduces a novel renormalisation group method combined with replica techniques to establish universality of Wigner's law for dependent entries.

Findings

Wigner's semicircle law holds under specific cumulant scaling bounds.

The approach applies to matrices with dependent entries.

The method connects renormalisation group ideas with random matrix universality.

Abstract

We show that if the non Gaussian part of the cumulants of a random matrix model obey some scaling bounds in the size of the matrix, then Wigner's semicircle law holds. This result is derived using the replica technique and an analogue of the renormalisation group equation for the replica effective action. This is a transcript of a talk given at "5th Winter Workshop on Non-Perturbative Quantum Field Theory" Sophia-Antipolis, March 2017 and is based on former work in collaboration with A. Tanasa and D.L. Vu, see arXiv:1609.01873 .