Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation
Pranay Agarwal, Riddhipratim Basu
TL;DR
The paper analyzes sharp tail bounds for transversal fluctuations in exponential last passage percolation on the plane, focusing on point-to-line geodesics and, at the halfway point, point-to-point geodesics. It develops a multi-segment path-construction framework and leverages BK-type inequalities to bound the probability of atypically large transversal fluctuations, obtaining a lower bound of $\mathbb{P}(|\bar{\Gamma}_n| \ge 2t(2n)^{2/3}) \ge C e^{-\frac{4}{3}t^3(1+o(1))}$ for the point-to-line case and a corresponding bound for the halfway-point of point-to-point geodesics that yields the exponent $\frac{8}{3}$ at $\gamma=1/2$. The results substantiate a conjecture of Liu (PTRF, 2022) in the symmetric case and provide a complementary perspective to the conjectured tail form $n^{2/3}p_{n,1/2}(t)=\exp(- (\frac{8}{3}+o(1))t^3)$. Overall, the work delivers sharp large-fluctuation tails and demonstrates that the required large deviation paths can be engineered through a controlled concatenation of geodesics with carefully balanced energy-entropy costs.
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).