Random matrices, Virasoro algebras, and noncommutative KP

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dc.creatorAdler, M.
dc.creatorShiota, T.
dc.creatorvan Moerbeke, P.
dc.date1998-12-03
dc.date.accessioned2026-07-07T06:17:42Z
dc.date.available2026-07-07T06:17:42Z
dc.descriptionWhat is the connection of random matrices with integrable systems? Is this connection really useful? The answer to these questions leads to a new and unifying approach to the theory of random matrices. 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 n eigenvalues in infinitesimal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalue in a Borel subset E); the latter probability is a ratio of tau-functions for the KP-equation, whose numerator satisfy partial differential equations, which decouple into the sum of two parts: a Virasoro-like part depending on time only and a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting t=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE's lead to a hierarchy of non-linear PDE's, purely in terms of the A_i. These PDE's are nothing else but the KP hierarchy for which the t-partials, viewed as commuting operators, are replaced by non-commuting operators in the endpoints A_i of the E under consideration. When the boundary of E consists of one point and for the known kernels, one recovers the Painleve equations, found in prior work on the subject.
dc.description56 pages
dc.identifierhttps://arxiv.org/abs/solv-int/9812006
dc.identifierhttp://arxiv.org/abs/solv-int/9812006
dc.identifierDuke Math Journal, 94, pp. 379-431, 1998
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/94475
dc.subjectExactly Solvable and Integrable Systems
dc.titleRandom matrices, Virasoro algebras, and noncommutative KP
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