Brownian motion and random matrices

This workshop, sponsored by AIM and NSF, was devoted to β-generalizations of the classical ensembles in random matrix theory. Recent advances have put stochastic methods on center stage, thus explaining the workshop title ‘Brownian motion and random matrices’. One recalls that a viewpoint on classical random matrix theory, generalizing Dyson’s three fold way, is that physically relevant ensembles are specified by the Hermitian part of the ten infinite families of matrix Lie algebras. By specifying a Gaussian weight, in each case the corresponding eigenvalue probability density function can be identified with the Boltzmann factor of a classical gas interacting via a repulsive pairwise logarithmic potential (log-gas). Curiously, the dimensionless inverse temperature β in this analogy is restricted to one of three values β = 1,2 or 4. This analogy, with the same restriction on β, carries over to the classical ensembles of random unitary matrices based on the ten families of symmetric spaces in correspondence with the matrix Lie algebras. In work dating from the first half of the previous decade, explicit constructions were given of random matrix ensembles with eigenvalue probability density functions realizing the log-gas Boltzmann factors for general β > 0. In the cases of the classical Gaussian Hermitian and circular ensembles, these construction are in terms of certain tridiagonal and unitary Hessenberg matrices respectively. Alternatively it was shown that the β-ensembles could be realized by certain families of random matrices defined recursively. In the second half of the the previous decade it was shown that the tri(and bi-) diagonal matrices appearing in the construction could be viewed as discretizations of certain differential operators perturbed by a noise term involving Brownian motion. Similarly, by analyzing the recurrences satisfied by the characteristic polynomials associated with the tridiagonal and unitary Hessenberg matrices, stochastic differential matrices were derived for the charaterization of the number of eigenvalues in a given interval in the bulk. This in turn was used to solve some previously intractable problems in random matrix theory, an example being the large distance asymptotic expansion of the spacing distributions for general β. The AIM workshop ‘Brownian motion and random matrices’ sought to build on these advances, and to tackle other problems of fundamental importance to random matrix theory. Three classes of problems were so identified.