21 April 2015 – Uzy Smilansky (Weizmann)

Title: Dyson’s Brownian Motion Model for Random Matrix Theory – Revisited

Abstract:  In his 1962 paper, F. Dyson introduced a then novel approach for studying random matrix ensembles in terms of Brownian dynamics in the space of matrices.

He then proposed a Fokker-Planck evolution for the spectral distribution function, whose stationary solution provides the spectral join probability distribution function, $P(\lambda_1, …,\lambda_N)$.

Here, we reformulate the approach for the traces, $t_n = \sum_{k=1}^{N} \lambda_{k}^{n}$  (= spectral moments), and derive the Fokker-Planck equations and their joint probability distribution $Q(t_1, …,t_N)$. The advantages of this version of Dyson’s theory will be discussed, and a few new identities between symmetric polynomials will be derived.