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

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.