The spectrum of coupled random matrices. (English) Zbl 0936.15018

Adler, M.; van Moerbeke, P.

Ann. Math. (2) 149, No. 3, 921-976 (1999).

The authors explain “how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitian random matrices and the equations the associated probabilities satisfy”.
In the Gaussian case the distribution of \(M_i\), \(i=1,2\), is proportional to \[ \exp\bigl \{-T_r (M^2_1+ M^2_2-2cM_1M_2) \bigr\} \] where the \(M_i\) are \(n\times n\) matrices.
The topics described involve bi-orthogonal polynomials, two-Toda lattice, vertex operators, Virasoro algebra and many others.

Reviewer: Alexei Khorunzhy (Khar’kov)

MSC:

15B52	Random matrices (algebraic aspects)
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory
17B68	Virasoro and related algebras
17B69	Vertex operators; vertex operator algebras and related structures

Keywords:

spectrum; Hermitian random matrices; bi-orthogonal polynomials; two-Toda lattice; vertex operators; Virasoro algebra