LA Probability Forum

UCLA, Math Sciences Bldg., Room 6627

In KPZ universality, an important family of models arises from 2D last-passage percolation (LPP): in a 2D i.i.d. random field, one considers the geodesic connecting two vertices, which is defined as the up-right path maximizing its weight, i.e., the sum/integral of the random field along it. A characteristic KPZ behavior is the 2/3 geodesic fluctuation exponent, which has been proven for some LPPs with exactly solvable structures. A topic of much recent interest is such models under upper- and lower-tail large deviations, i.e., when the geodesic weight is atypically large or small. In prior works, it was established that the geodesic exponent changes to 1/2 (more localized) and 1 (delocalized) respectively. In this talk, I will describe a further refined picture: the geodesic converges to a Brownian bridge under the upper tail, and a uniformly chosen function from a one-parameter family under the lower tail. I will also discuss the proofs, using a combination of algebraic, geometric, and probabilistic arguments.

This is based on one forthcoming work with Shirshendu Ganguly and Milind Hegde, and one ongoing work with Shirshendu Ganguly.