The Spectrum of Coupled Random Matrices

In this work, we explain how the integrable technology can be brought to bear to gain insight in the nature of the distribution of the spectrum of coupled Hermitean random matrices (of finite size) and the equations the associated probabilities satisfy. Given two intervals, the joint statistics random matrices satisfy very simple non-linear third-order partial differential equations in the end points of the intervals E₁ and E₂; these equations are independent of the size n of the matrices. This is based on the 2-Toda lattice, its algebra of symmetries and its vertex operators. Namely, the method is to introduce time parameters, in an artificial way, and to "{\ dress up}" a certain matrix integral with a "{\ vertex integral operator}", for which we find Virasoro-like differential equations. Combining these equations with a new partial differential equation for the two-Toda Lattice \τ-functions leads to that result. In the course of doing this, we also give the Virasoro constraints for certain matrix integrals with arbitrary boundaries.