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• W2012238557 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=1r [A2i−1, A2i], in terms of the endpoints Ai. 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 tj and the endpoints Ai of the intervals J. The Virasoro vector fields on τ move the endpoints (motion in moduli space) according to the simple (decoupled) differential equations Ȧi = Aik+1." @default.
• W2012238557 created "2016-06-24" @default.
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• W2012238557 date "1995-11-01" @default.
• W2012238557 modified "2023-09-26" @default.
• W2012238557 title "Random matrices, vertex operators and the Virasoro algebra" @default.
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• W2012238557 doi "https://doi.org/10.1016/0375-9601(95)00725-i" @default.
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