  The study of the spectrum of coupled random matrices has received rather
little attention. To the best of our knowledge, coupled random matrices have
been studied, to some extent, by Mehta. In this work, we explain how the
integrable technology can be brought to bear to gain insight into the nature of
the distribution of the spectrum of coupled Hermitean random matrices and the
equations the associated probabilities satisfy. In particular, the two-Toda
lattice, its algebra of symmetries and its vertex operators will play a
prominent role in this interaction. 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. These methods lead to very simple nonlinear third-order partial
differential equations for the joint statistics of the spectra of two coupled
Gaussian random matrices.
