Harriet Walsh

Random growth in half space and solutions of Painlevé equations

I will talk about a model of two dimensional random growth (namely, polynuclear growth) which can be translated into a probability law on integer partitions (by way of the RSK algorithm). We can find exact expressions for statistics of this model with algebraic tools, and compute fine asymptotics. I will focus on a model in half space with external sources driving growth at the edges. The limiting distribution of interface fluctuations in this model interpolates between different universal Tracy—Widom distributions from random matrix theory, and encodes solutions of the Painlevé II differential equation. At one point it matches a half-space version of the Baik—Rains distribution found by Barraquand, Krajenbrink and Le Doussal using methods from physics. Our approach uses connections between symmetric functions, matrix integrals, and Hankel determinants, plus a Riemann—Hilbert problem. Based on joint work with Mattia Cafasso, Alessandra Occelli and Daniel Ofner.