Coalescence of geodesics in exactly solvable models of last passage percolation

Abstract

Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R−c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (−k2/3, k2/3) and v4 = (k2/3, −k2/3), we establish the optimal tail estimate P(|v|1>Rk)≍R−2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187–3206 (2016)] who proved the corresponding lower bound.