The spectrum of coupled random matrices

0. Introduction 1. Operators Λ and ε with [Λ, ε] = 1 and the δ-function 2. The two-Toda lattice 3. Bilinear Fay identities and a new identity for the two-Toda τ -functions 4. Higher Fay identities for the two-Toda lattice 5. Eigenfunction expansions and vertex operators 6. A remarkable trace formula 7. Two-Toda symmetries and the ASV-correspondence 8. Fredholm determinants of Christoffel-Darboux kernels 9. Differential equations for vertex operators and a Virasoro algebra of central charge c = −2 10. Vertex representation of probabilities and Virasoro constraints 11. PDE’s for the joint statistics of the spectra of Gaussian coupled random matrices 12. Coupled random matrices with the Laguerre statistics