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.
Citation
Lingfu Zhang. "Optimal exponent for coalescence of finite geodesics in exponential last passage percolation." Electron. Commun. Probab. 25 1 - 14, 2020. https://doi.org/10.1214/20-ECP354
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