Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation

2024/05/28 by Pranay Agarwal, Agarwal, Pranay, Riddhipratim Basu +1
Mathematics · Computer Science · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Bayesian Methods and Mixture Models

paper · pdf · doi:10.48550/arxiv.2405.18056

Abstract

For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that p^*_n(t), the probability that the point-to-line geodesic from the origin to the line x+y=2n ends at (n-t(2n)^2/3, n+t(2n)^2/3) satisfies that n^2/3p^*_n(t)=exp(-((4)/(3)+o(1))t³) for t large and p_n,(1)/(2)(t), the probability that the geodesic from the origin to the point (n,n) passes through the point ((1)/(2)n-tn^2/3, (1)/(2) n+tn^2/3), satisfies n^2/3p_n,(1)/(2)(t)=exp(-((8)/(3)+o(1))t³) for t large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022).

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