Random matrices, vertex operators and the Virasoro algebra

Abstract

This work aims at a new approach to the theory of random matrices, inspired by recent work on matrix models. Introducing an appropriate time t-dependence in the probability distribution of the matrix ensemble leads to vertex operator expressions for the n-point correlation functions (probabilities of an eigenvalue in infinitesimal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalue in an interval); the latter satisfy Virasoro-like constraints, which, upon setting t = 0, lead to a new hierarchy of PDEs for the P (no eigenvalue ∈ J), where J = ∪ i=1 r [ A 2 i−1 , A 2 i ], in terms of the endpoints A i . In the single interval case, the first equation in the hierarchy recovers the Painlevé distributions for the classical ensembles. This is done in Section 1 for polynomial ensembles, i.e., the probabilities are given by explicit matrix integrals, and in Section 3 for ensembles, defined by more general kernels. Examples are given in Section 4. From the point of view of the KP and Toda symmetries and their Virasoro (or W)-counterparts on τ, as studied by us previously, the probabilities above are expressed in terms of a τ-function τ( t, A), depending on the integrable directions t j and the endpoints A i of the intervals J. The Virasoro vector fields on τ move the endpoints (motion in moduli space) according to the simple (decoupled) differential equations A ̇ i = A i k+1 .