Student Probability/PDE Seminar: The Weight, Geometry and Coalescence of Scaled Polymers in Brownian Last Passage Percolation

Seminar | September 8 | 2:10-3:30 p.m. | 891 Evans Hall

 Alan Hammond, UC Berkeley

 Department of Mathematics

In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice consisting of independent and identically distributed weights is considered. The weight of an upright path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation. We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression. Building on a study arXiv:1609.02971 concerning the decay in probability for the existence of several near polymers with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in scaled coordinates has a superpolynomial decay rate in k. This result has implications for the Brownian regularity of the scaled polymer weight profile begun from rather general initial data.