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.