Large Deviations for Geodesic Midpoint Fluctuations in Last Passage Percolation

Abstract

Last passage percolation is a representative model of phenomena that characterize the Kardar-Parisi-Zhang universality class. The model has an interesting integrable structure that allows for the computation of many surprising exact formulas, based on connections with random matrices, representation theory, orthogonal polynomials, and more. The last 5-6 years has even seen the construction of LPP’s universal scaling limit, called the directed landscape. This talk focuses on the pre-limiting lattice model, specifically on large deviations for its geodesic paths. As for simple random walk, the large deviations formulas are not universal. Instead they depend on the distribution of the iid inputs, which is an aspect of the general problem that is much less studied. I will describe our characterization of these large deviations formulas in terms of last passage times. For integrable choices of the iid weights these characterizations lead to very explicit formulas, which allow us to directly compare the large deviations for the geodesic to those of random walk paths. Based on joint work with Riddhi Basu, Sean Groathouse, and Xiao Shen.