Dyson's Brownian-motion model for random matrix theory - revisited. With an Appendix by Don Zagier

Christopher H. Joyner, Uzy Smilansky

Published 2015-03-22Version 1

We offer an alternative viewpoint on Dyson's original paper regarding the application of Brownian motion to random matrix theory (RMT). In particular we show how one may use the same approach in order to study the stochastic motion in the space of matrix traces $t_n = \sum_{\nu=1}^{N} \lambda_\nu^n$, rather than the eigenvalues $\lambda_\nu$. In complete analogy with Dyson we obtain a Fokker-Planck equation that exhibits a stationary solution corresponding to the joint probability density function in the space $t = (t_1,\ldots,t_n)$, which can in turn be related to the eigenvalues $\lambda = (\lambda_1,\ldots,\lambda_N)$. As a consequence two interesting combinatorial identities emerge, which are proved algebraically in the appendix. We also offer a number of comments on this version of Dyson's theory and discuss its potential advantages.