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

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.

References in corpus (11)

• Shape Fluctuations and Random Matrices
• Competition interfaces and second class particles
• Airy processes and variational problems
• Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem
• Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• Bigeodesics in first-passage percolation
• Random coalescing geodesics in first-passage percolation
• Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation
• Interlacing and scaling exponents for the geodesic watermelon in last passage percolation
• Lower Deviations in $\beta$-ensembles and Law of Iterated Logarithm in Last Passage Percolation

Cited by in corpus (5)

• Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• On the exponent governing the correlation decay of the Airy${}_1$ process
• Mixing times and cutoff for the TASEP in the high and low density phase
• Geodesic trees in last passage percolation and some related problems