This week's seminars
Statistics Seminars: Nonexistence of bi-infinite geodesics in exponential last passage percolation - a probabilistic way (Joint work with Ofer Busani and Timo Seppäläinen)
11 November 2019 13:00 in CM107
Take a point on the 2-dimensional integer lattice and another one North-East from the first. Place i.i.d. Exponential weights on the vertices of the lattice; the point-to-point geodesic between the two points is the a.s. unique path of North and East steps that collects the maximal sum of these weights.
A bi-infinite geodesic is a doubly infinite North-East path such that any segment between two of its points is a point-to-point geodesic. We show that this thing a.s. does not exist (except for the trivial case of the coordinate axes). The intuition is roughly this: transversal fluctuations of a point-to-point geodesic are in the order of the 2/3rd power of its length, which becomes infinite for a bi-infinite geodesic. This and coalescence of geodesics result in not seeing this path anywhere near the origin which, combined with translation invariance, a.s. excludes its existence.
One needs to make this more quantitative to prove that even after taking the union for all possible directions we cannot see a bi-infinite geodesic, a program sketched by Newman. This has recently been completed rigorously by Basu, Hoffman and Sly with inputs from integrable probability. In this work we instead build on purely probabilistic arguments, such as couplings and maxima of drifted random walks, to arrive to this result.
Contact firstname.lastname@example.org for more information