## Electronic Communications in Probability

### Optimal exponent for coalescence of finite geodesics in exponential last passage percolation

Lingfu Zhang

#### 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 directed paths from vertices $(0,\lfloor k^{2/3}\rfloor )$ and $(\lfloor k^{2/3} \rfloor ,0)$ to $(n,n)$, respectively. For the coalescence point of these paths, we show that the probability for it being $>Rk$ 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 [7], 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.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 74, 14 pp.

Dates
Accepted: 5 October 2020
First available in Project Euclid: 22 October 2020

https://projecteuclid.org/euclid.ecp/1603332091

Digital Object Identifier
doi:10.1214/20-ECP354

#### Citation

Zhang, Lingfu. Optimal exponent for coalescence of finite geodesics in exponential last passage percolation. Electron. Commun. Probab. 25 (2020), paper no. 74, 14 pp. doi:10.1214/20-ECP354. https://projecteuclid.org/euclid.ecp/1603332091

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