F Metz Blog Article December 2017

New Mathematical Approach to Quantify Eigenvalue Fluctuations in Random Matrices

Since the pioneering work of Wigner in the 50’s, random matrix theory has been playing a crucial role in the understanding of several problems in physics, such as the statistics of nuclear energies, quantum chaos and disordered electronic systems. These are traditional examples where the theory of Gaussian random matrices has achieved a remarkable success. The general strategy goes as follows: an intricate physical system with many interacting entities is represented by a matrix, whose elements are drawn from a Gaussian probability distribution.

Several mathematical techniques are particularly suited to deal with these Gaussian random matrices and, as a consequence, a plethora of information about their statistical behaviour is available. Much less is known about non-Gaussian random matrices, where the matrix entries simply do not follow from a Gaussian probability. Thanks to the increasing capacity of computers in processing large amounts of data, non-Gaussian random matrices have become more popular in the last 20 years, since the structure of network systems (Internet, power-grids, transportation networks, neural networks, etc.) can be pictured as a random matrix. The usual mathematical tools of random matrix theory, well-adapted to handle Gaussian matrices, fail in describing the statistical properties of non-Gaussian random matrices. In this paper, LML External Fellow Fernando Metz introduces a novel mathematical approach to study the statistical behaviour of a certain class of non-Gaussian random matrices. The advantages and limitations of this new technique are tested by considering its performance on Gaussian matrices, for which many reliable results are available. This work paves the way to study non-Gaussian random matrices in an analytic way.

 

Journal of Physics A: Mathematical and Theoretical Volume 50 Number 49

Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices

Published 16 November 2017 © 2017 IOP Publishing Ltd

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