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## The Airy and Pearcey Processes

### Craig A. Tracy

U.C. Davis

###
**Friday**, January 14, 2005

60 **Carson** Hall, 3 pm

In honor of Laurie Snell's 80th Birthday

**Abstract: ** The *extended Airy kernel* describes the
space-time correlation functions for the *Airy process*, which
is the limiting process for a polynuclear growth model. The Airy
functions themselves are given by integrals in which the exponents
have a cubic singularity, arising from the coalescence of two saddle
points in an asymptotic analysis. Pearcey functions are given by
integrals in which the exponents have a quartic singularity, arising
from the coalescence of three saddle points. Corresponding
*Pearcey kernels* appear in random matrix theory and a Brownian
motion model for a fixed time. We derive an *extended Pearcey
kernel* by scaling the Brownian motion model at several times, and a
system of partial differential equations whose solution determines
associated distribution functions. We expect, but have not proved,
that there is a limiting nonstationary process, consisting of
infinitely many paths, whose space-time correlations are expressible
in terms of this extended kernel. These recent developments will be
surveyed. This is joint work with Harold Widom.

This talk will be accessible to graduate students.