Optimal Exponent for Coalescence of Finite Geodesics in Exponential Last Passage Percolation

TL;DR

This paper investigates the coalescence behavior of finite geodesics in exponential last passage percolation, establishing the optimal decay exponent for the probability that the coalescence point is far from the origin.

Contribution

It determines the precise decay exponent for the coalescence point distance in finite geodesics, extending previous semi-infinite results to finite paths in exactly solvable models.

Findings

Probability decays as R^{-2/3} for coalescence point being >kR away

Results apply to other exactly solvable last passage percolation models

Provides the optimal exponent for finite geodesic coalescence

Abstract

In this note, we study the model of directed last passage percolation on $\mathbb{Z}^2$, with i.i.d. exponential weight. We consider the maximum paths from vertices $\left(0,\lfloor k^{2/3} \rfloor\right)$ and $(\lfloor k^{2/3} \rfloor,0)$ to $(n,n)$, respectively. For the coalescing point of these paths, we show that the probability for it being $>kR$ far away from the origin is in the order of $R^{-2/3}$. This is motivated by a recent work of Basu, Sarkar, and Sly, where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open. Our arguments also apply to other exactly solvable models of last passage percolation.