# The Distribution of the first Eigenvalue Spacing at the Hard Edge of Random Hermitian Matrices

#### by N. S. Witte

Institution: Department of Mathematics and Statistics, The University of Melbourne
Date: Thu 26th May 2005
Time: 3:15 PM
Location: Room 213,Richard Berry Building

Abstract: The distribution for the first eigenvalue spacing at the finite
lower endpoint of the spectrum of the Laguerre unitary ensemble of finite
rank random matrices is found in terms of the fifth Painlev\'e system and
the solution of it's associated linear isomonodromic system. In particular
it is characterised by the polynomial solutions to the isomonodromic
equations which are also orthogonal with respect to a deformation of the
Laguerre weight. In the scaling to the hard edge regime we find an analogous
situation where a certain Painlev\'e III system and it's associated
linear isomonodromic system characterise the scaled distribution. We
undertake extensive analytical studies of this system and use this
knowledge to accurately compute the distribution and its moments.

(joint work with P.J. Forrester)