24.04.2007 (Tuesday)

Universality in critical unitary random matrix ensembles and Painleve equations

Regular Seminar Tom Claeys (Leuven)

16:00 Brunel U.
room M128

The eigenvalues of large unitary random matrices show universal local behavior, only depending on the so-called scaling regime. It is known that in the bulk of the spectrum, local correlations of the eigenvalues are given in terms of the sine kernel, while at the edge one obtains the Airy kernel. In certain critical random matrix ensembles, other limiting correlation kernels can occur. Near singular interior points, the limiting kernel is related to the Hastings-McLeod solution of the Painleve II equation. Near singular edge points on the other hand, one obtains a kernel related to the second member of the Painleve I hierarchy. We describe how one can obtain these kernels rigorously in double scaling limits using the Riemann-Hilbert approach.