Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1998-12-03
Duke Math Journal, 94, pp. 379-431, 1998
Nonlinear Sciences
Exactly Solvable and Integrable Systems
56 pages
Scientific paper
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.
Adler Mark
Moerbeke Pierre van
Shiota Takahiro
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