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 $\mathbb{P}(|v|_1> Rk) \leq R^{-c}$ 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.