WHAT IS... RANDOM MATRIX THEORY

Random matrix theory is the study of matrix-valued random variables, or, slightly changing the perspective, measures on spaces of matrices. This theory dates back to Wishart's work in statistics in 1928 (5), followed by Wigner's study of the distribution of nuclear energy levels in the 50's (4). The formulation of the problems in the theory, by using Haar mesure of some particular ensemble of matrices, amounts to study the repartition of the eigenvalues of the matrix. One of the reasons for its success is universality : the eigenvalues correlation correctly rescaled do not depend on the probability distribution. This property, which has been observed for the Riemann Zeta function by Montgomery (3) and for a special ensemble of random matrices by Dyson, laid the foundations for new hypotheses concerning the repartition of primes, until the consideration by Keating and Snaith (2) that the Riemann Zeta function can be modeled by the characteristic polynomial of a particular type of random matrices, revealing an unexpected link with the Hilbert-Polya conjecture.