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

TL;DR

This paper derives sharp deviation bounds for the transversal fluctuations of geodesics in exponential last passage percolation, quantifying the probabilities of large deviations at endpoints and midpoints.

Contribution

It provides explicit exponential decay rates for the probabilities of large transversal fluctuations, confirming a special case of Liu's conjecture from 2022.

Findings

Probability of large endpoint fluctuation scales as exp(- (4/3) t^3)

Probability of large midpoint fluctuation scales as exp(- (8/3) t^3)

Results verify a conjecture in last passage percolation theory

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 $n^{2/3}p^*_n(t)=\exp(-(\frac{4}{3}+o(1))t^{3})$ for $t$ large and $p_{n,\frac{1}{2}}(t)$, the probability that the geodesic from the origin to the point $(n,n)$ passes through the point $(\frac{1}{2}n-tn^{2/3}, \frac{1}{2} n+tn^{2/3})$, satisfies $n^{2/3}p_{n,\frac{1}{2}}(t)=\exp(-(\frac{8}{3}+o(1))t^3)$ for $t$ large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022).