The spectrum of coupled random matrices

M. Adler and P. van Moerbeke

Review from Zentralblatt MATH:

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.

Reviewed by Alexei Khorunzhy

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

Classification (MSC2000): 15A52 81R50 17B68 17B69

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