# Probability and Mathematical Physics Seminar

#### Geometry of geodesics through Busemann measures in directed last-passage percolation

**Speaker:**
Firas Rassoul-Agha, University of Utah

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, September 13, 2019, 11:10 a.m.

**Synopsis:**

We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and describe the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The main tool is the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model we give a complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics. Part of our results concerns the existence of exceptional (random) directions in which new interesting shock structures occur.

Joint work with Chris Janjigian and Timo Seppalainen