The spectrum of random matrices and integrable systems

Abstract

What is the connection of random matrices with integrable systems? Is this connection really useful? Introducing apprpriate times in the distribution of the ensemble of matrices, one shows that the corresponding distribution of the eigenvalues satisfies the KP-equation, the 1-Toda lattice or the 2-Toda lattice, depending on the original distribution. The probability distribution also satisfies Virasoro type constraints, which contain a time-part and a boundary-part. These equations taken together lead to a system of PDE's for the distribution of the spectrum in terms of the boundary of the set, under consideration.

17 pages, Latex, group21.sty

Cited by in corpus (5)

• Partition functions for Matrix Models and Isomonodromic Tau functions
• Geometry of Spectral Curves and All Order Dispersive Integrable System
• Integrable Lattices: Random Matrices and Random Permutations
• Symmetric random matrices and the Pfaff lattice
• Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum