The Geometry of the Last Passage Percolation Problem

Abstract

Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) and I have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model and our work to use it to produce probabilistic information.

Date
Apr 24, 2017 16:50 +0100 — Apr 28, 2017 17:35 +0100
Location
Centre International de Rencontres Mathematiques
163 Avenue de Luminy, Case 916, Marseille, M5T 3J1
Tom Alberts
Tom Alberts
Associate Professor of Mathematics
University of Utah