The spectrum of random matrices and integrable systems

Details The spectrum of random matrices and integrable systems The spectrum of random matrices and integrable systems

1997-06-25

arxiv.org/abs/solv-int/9706009v1

Group21, Physical applications and Mathematical aspects of Geometry, Groups and Algebras, Vol II, 835--852, Eds.: Doebner, Sch

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages, Latex, group21.sty

Scientific paper

What is the connection of random matrices with integrable systems? Is this connection really useful? Introducing apprpriate times in the distribution of the ensemble of matrices, one shows that the corresponding distribution of the eigenvalues satisfies the KP-equation, the 1-Toda lattice or the 2-Toda lattice, depending on the original distribution. The probability distribution also satisfies Virasoro type constraints, which contain a time-part and a boundary-part. These equations taken together lead to a system of PDE's for the distribution of the spectrum in terms of the boundary of the set, under consideration.

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