Wigner law for matrices with dependent entries - a perturbative approach

TL;DR

This paper demonstrates that the Wigner semi-circle law applies to Hermitian matrices with dependent entries under certain cumulant deviation bounds, using a quantum field theoretical approach with renormalisation group techniques.

Contribution

It introduces a perturbative method to extend the Wigner law to matrices with dependent entries via a quantum field theory framework.

Findings

Wigner law holds under specific cumulant deviation bounds

Effective potential satisfies a renormalisation group equation

Perturbative approach applicable to dependent entries matrices

Abstract

We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we use replicas interpreted as a zero-dimensional quantum field theoretical model whose effective potential obey a renormalisation group equation.