dorsal/arxiv
View SchemaRandom matrices, Virasoro algebras, and noncommutative KP
| Authors | M. Adler, T. Shiota, P. van Moerbeke |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9812006 |
| URL | https://arxiv.org/abs/solv-int/9812006 |
| Journal | Duke Math Journal, 94, pp. 379-431, 1998 |
Abstract
What 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.
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"abstract": "What is the connection of random matrices with integrable systems? Is this\nconnection really useful? The answer to these questions leads to a new and\nunifying approach to the theory of random matrices. Introducing an appropriate\ntime t-dependence in the probability distribution of the matrix ensemble, leads\nto vertex operator expressions for the n-point correlation functions\n(probabilities of n eigenvalues in infinitesimal intervals) and the\ncorresponding Fredholm determinants (probabilities of no eigenvalue in a Borel\nsubset E); the latter probability is a ratio of tau-functions for the\nKP-equation, whose numerator satisfy partial differential equations, which\ndecouple into the sum of two parts: a Virasoro-like part depending on time only\nand a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting\nt=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE\u0027s lead to\na hierarchy of non-linear PDE\u0027s, purely in terms of the A_i. These PDE\u0027s are\nnothing else but the KP hierarchy for which the t-partials, viewed as commuting\noperators, are replaced by non-commuting operators in the endpoints A_i of the\nE under consideration. When the boundary of E consists of one point and for the\nknown kernels, one recovers the Painleve equations, found in prior work on the\nsubject.",
"arxiv_id": "solv-int/9812006",
"authors": [
"M. Adler",
"T. Shiota",
"P. van Moerbeke"
],
"categories": [
"solv-int",
"nlin.SI"
],
"journal_ref": "Duke Math Journal, 94, pp. 379-431, 1998",
"title": "Random matrices, Virasoro algebras, and noncommutative KP",
"url": "https://arxiv.org/abs/solv-int/9812006"
},
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