Harriet Walsh
Stationary random growth in half space, and connections
I will talk about a model of two dimensional random growth (namely, polynuclear growth) where we can find nice exact expressions for the distributions of key statistics. By analysing this model in half-space with external sources, we can show the appearance of a universal interface fluctuations associated with stationary random growth, previously studied by Beta, Ferrari and Occelli, and then Barraquand, Le Doussal and Krajenbrink. We also find a distribution which interpolates between the half-space stationary one and different Tracy-Widom distributions from random matrix theory. Our approach uses connections between symmetric functions, matrix integrals, and Hankel determinants, plus a Riemann-Hilbert problem. I’ll discuss how we can extend this approach to an inhomogeneous version of TASEP. Based on joint work with Mattia Cafasso, Alessandra Occelli and Daniel Ofner.
James Martin
Interchangeability of rates in interacting particle systems and last-passage percolation
There are many examples of inhomogeneous exactly-solvable interacting particle systems (or queueing networks, or percolation models, ...) with interesting invariance properties under permutation of their rates. I will describe some old and new results and some applications.
Oleg Zaboronski
Entrance laws for coalescing and annihilating Brownian motions
Systems of instantaneously annihilating or coalescing Brownian motions on the line are considered. The extreme points of the set of entrance laws for this process are shown to be Pfaffian point processes at all times and their kernels are identified. (Joint work with Roger Tribe).
Will FitzGerald
Ordered random walks and the Airy Line Ensemble
The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang (KPZ) universality class. In the KPZ universality class, it is related to the universal limiting objects that describe the scaling limit of a large class of random interface growth models and interacting particle systems. In random matrix theory, it is the edge scaling limit of Dyson Brownian motion, the evolution of the eigenvalues of Brownian motion in the space of Hermitian matrices. I will discuss these connections and a universality property of the Airy line ensemble. Consider a growing number of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. The top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power of the expected number of random walk steps. Based on joint work with Denis Denisov and Vitali Wachtel.