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

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.


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
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