Introduction to Random Matrices
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چکیده
Here I = [-Ji (a2j-l,a2J) and XI(Y) is the characteristic function of the set I . In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is equal to v(a). Also v(a) is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the aj's a r e the independent variables) that were first derived by Jimbo, Miwa, M6ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev~ V equation. For large s we give an asymptotic formula for E2(n; s), which is the probability in the GUE that exactly n eigenvalues lie in an interval of length s.
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تاریخ انتشار 1993