Malte Hassler (University of Edinburgh) – Coalescence of finite and infinite geodesics in general last passage percolation

In last passage percolation, we independently put weights on the vertices of the square lattice according to some probability distribution. We then study the up-right paths between points on the lattice that collect as much weight as possible. A phenomenon known as KPZ universality postulates that the essential behavior of the maximum weight and the corresponding optimal paths does not depend on the choice of weight distribution (subject to a few regularity conditions). So far however, this can only be verified for geometric and exponential distributions. In this talk, we look at general continuous distributions with finite exponential moments and prove upper probability bounds on various events where the weight maximizing paths coalesce.

More precisely, we show the following under a uniform curvature assumption: Locally for a deterministic full-Lebesgue-measure set of directions, we give a polynomial upper probability bound for the event that a Busemann function exhibits a jump in an interval of directions. The proof of the result has three parts: First, we give a bound on the coalescence time of a point-to-point geodesic and an infinite geodesic going in the same direction and second, for two infinite geodesics of distinct directions. Third, we bound the coalescence time for two finite geodesics with nearby endpoints. The latter result is an adaptation of a theorem for first passage percolation by Dembin, Elboim and Peled. It does not require any assumptions on the limit shape and gives stronger bounds the closer the endpoints are to each other.