Advances in High Energy Physics

Volume 2017 (2017), Article ID 4098720, 7 pages

https://doi.org/10.1155/2017/4098720

## A Renormalisation Group Approach to the Universality of Wigner’s Semicircle Law for Random Matrices with Dependent Entries

Centre de Physique Théorique, Aix-Marseille Université, Marseille, France

Correspondence should be addressed to Thomas Krajewski

Received 15 June 2017; Revised 1 September 2017; Accepted 1 October 2017; Published 31 December 2017

Academic Editor: Ralf Hofmann

Copyright © 2017 Thomas Krajewski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We show that if the non-Gaussian part of the cumulants of a random matrix model obeys 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.

#### 1. Introduction

Random matrix theory (see the classical text [1]) first appeared in physics in Wigner’s work on the level spacing in large nuclei. Since then, it has proven to have multiple applications to physics and other branches of science (see, e.g., [2]). Most of these applications rely on the universal behaviour of some of the observables for matrices of large size. A simple example is Wigner’s semicircle law for the eigenvalue density that holds in the large limit for matrices whose entries are independent and identically distributed.

Understanding the universal behaviour of eigenvalue distributions and correlations ranks among the major problems in random matrix theory. In this respect, the renormalisation group turns out to be a powerful technique. Introduced in the context of critical phenomena in statistical mechanics by K. Wilson to account for the universality of critical exponents, the latter has also been proven to be useful in understanding probability theory. For instance, it leads to an insightful proof of the central limit theorem (see the review by Jona-Lasinio [3] and references therein).

The renormalisation group has been used to derive the semicircle law for random matrices in the pioneering work of Brézin and Zee [4]. In the latter approach, the renormalisation group transformation consists in integrating over the last line and column of a matrix of size to reduce it to a matrix of size . This leads to a differential equation for the resolvent in the large limit whose solution yields the semicircle law.

In this paper, we follow a different route: we first express the resolvent as an integral over replicas and introduce a differential equation for the replica effective action. This differential equation is a very simple analogue of Polchinski’s exact renormalisation group equation [5]. It is used to derive inductive bounds on the various terms, ensuring that the semicircle law is obeyed provided the cumulants of the original matrix model fulfil some simple scaling bounds in the large limit.

This paper is based on some work in collaboration with Krajewski et al. in which we extend Wigner’s law to random matrices whose entries fail to be independent [6] to which we refer for further details. There have been other works on such an extension (see [7–9]).

#### 2. What Are Random Matrices?

A random matrix is a probability law on a space of matrices, usually given by the joint probability density on its entries: Thus, a random matrix of size is defined as a collection of random variables. However, there is a much richer structure than this, relying notably on the spectral properties of the matrices.

Here, we restrict our attention to a single random matrix. Note that it is also possible to consider several random matrices, in which case the noncommutative nature of matrix multiplication plays a fundamental role, leading to the theory of noncommutative probabilities.

There are two important classes of probability laws on matrices:(i)Wigner ensemble: the entries are all independent variables: up to the Hermitian condition .(ii)Unitary ensemble: the probability law is invariant under unitary transformations: for any unitary matrix .

The only probability laws that belong to both classes are the Gaussian ones: up to a shift of by a fixed scalar matrix.

The main objects of interest are the expectation values of observables, defined as Among the observables, the spectral observables defined as symmetric functions of the eigenvalues of play a crucial role in many applications. This is essentially due to their universal behaviour: in the large limit, for some matrix ensembles and in particular regimes, the expectation values of specific spectral observables do not depend on the details of the probability law .

Universality is at the root of the numerous applications to physics and other sciences, since the results we obtain are largely model-independent. Among the applications to physics, let us quote the statistics of energy levels in heavy nuclei, disordered mesoscopic systems, quantum chaos, chiral Dirac operators, and so forth.

#### 3. Wigner’s Semicircle Law

In this paper, we focus on the eigenvalue density, defined as In particular, a universal behaviour is expected in the large limit for some ensembles.

For a Gaussian random Hermitian matrix , the eigenvalue density obeys Wigner’s semicircle law: Empirically, may be determined by plotting the histogram of eigenvalue of a matrix taken at random with a given probability law (see Figure 1).