Random Matrices, Vertex Operators and the Virasoro-algebra

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 is an element of J), where J = boolean OR(i=1)(r) [A(2i-1), A(2i)], in terms of the endpoints A(i). In the single interval case, the first equation in the hierarchy recovers the Painleve 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 tau, as studied by us previously, the probabilities above are expressed in terms of a tau-function tau(t, A), depending on the integrable directions t(j) and the endpoints A(i) of the intervals J. The Virasoro vector fields on tau move the endpoints (motion in moduli space) according to the simple (decoupled) differential equations A(i) = A(i)(k+1).