A nonlinear equation and its application to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices

P.J. Forrester
Department of Mathematics
University of Melbourne
Parkville, Victoria 3052

Andrew M. Odlyzko
AT&T Bell Laboratories
Murray Hill, New Jersey

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A nonlinear equation generalizing the $\omega$ form of the Painlev V equation is used to compute the probability density function for the distance from an eigenvalue of a matrix from the GUE ensemble to the eigenvalue nearest to it. (The classical results concern distribution of the distances between consecutive eigenvalues.) Comparisons are made with the corresponding distribution for zeros of the Riemann zeta function, which are conjectured to behave like eigenvalues of large random GUE matrices.