I will talk about models of two dimensional random growth (namely, polynuclear growth) which can be translated into probability laws on integer partitions by way of the RSK algorithm. As a consequence, we can study their asymptotic statistics with algebraic tools. I will focus on a model in half space with external sources driving growth at the edges, and present a new asymptotic distribution governing its interface fluctuations which interpolates between different universal Tracy-Widom distributions from random matrix theory, and encodes solutions of the Painlevé II integrable differential equation. Our approach uses connections between symmetric functions, matrix integrals, Hankel determinants, and a Riemann-Hilbert problem. Based on joint work with Mattia Cafasso, Alessandra Occelli and Daniel Ofner.
