# 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)

*For More Information:* Iwan Jensen tel: 8344-5214 email: i.jensen@ms.unimelb.edu.au