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 ${\mathbb{Z}}^2$ with i.i.d. exponential weights on the vertices. Fix two points $v_1=(0,0)$ and $v_2=(0,\lfloor k^{2/3}\rfloor )$ for some $k>0$, and consider the maximal paths ${\Gamma }_1$ and ${\Gamma }_2$ starting at $v_1$ and $v_2$ respectively to the point $(n,n)$ for $n\gg k$. Our object of study is the point of coalescence, i.e., the point $v\in {\Gamma }_1\cap {\Gamma }_2$ with smallest $|v{|}_1$. We establish that the distance to coalescence $|v{|}_1$ scales as $k$, by showing the upper tail bound $\mathbb{P}(|v{|}_1>Rk)\le R^{-c}$ 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 $v_3=(-\lfloor k^{2/3}\rfloor ,\lfloor k^{2/3}\rfloor )$ and $v_4=(\lfloor k^{2/3}\rfloor ,-\lfloor k^{2/3}\rfloor )$, we establish the optimal tail estimate $\mathbb{P}(|v{|}_1>Rk)\asymp R^{-2/3}$, for the point of coalescence $v$. This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound.

arXiv comment30 pages, 7 figures. Optimal tail estimates for semi-infinite geodesics added in this version