Pierre van Moerbeke
(Universite de Louvain, Belgium)
A Partial differential equation for the Pearcey distribution
Brezin and Hikami studied a random matrix ensemble with external source,
where the support of the equilibrium measure consists of one interval, with
a gap in the middle. A fine tuning of the external source leads to a
closure of the gap at the origin. The statistics of the eigenvalues of the
problem near this critical point is described by the Fredholm determinant
of the Pearcey kernel. This lecture gives a fourth order non-linear PDE for
the probability that an interval contains no eigenvalues.