Electronic Communications in Probability

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

Lingfu Zhang

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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

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

Received: 18 March 2020
Accepted: 5 October 2020
First available in Project Euclid: 22 October 2020

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C23: Exactly solvable dynamic models [See also 37K60]

exponential last passage percolation coalescence of geodesics optimal exponent

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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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