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Introduction to Random Matrices | | Craig A. Tracy
; Harold Widom
; | | Date: |
14 Oct 1992 | | Journal: | Springer Lecture Notes in Physics 424 (1993) 103-130 | | Subject: | High Energy Physics - Theory; Exactly Solvable and Integrable Systems; Mathematical Physics | hep-th cond-mat math-ph math.MP nlin.SI solv-int | | Abstract: | These notes provide an introduction to the theory of random matrices. The central quantity studied is $ au(a)= det(1-K)$ where $K$ is the integral operator with kernel $1/pi} {sinpi(x-y)over x-y} chi_I(y)$. Here $I=igcup_j(a_{2j-1},a_{2j})$ and $chi_I(y)$ 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 $ au(a)$. Also $ au(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$’s are the independent variables) that were first derived by Jimbo, Miwa, M{ôri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{é V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$. | | Source: | arXiv, hep-th/9210073 | | Services: | Forum | Review | PDF | Favorites | |
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