Small deviation estimates and small ball probabilities for geodesics in last passage percolation

TL;DR

This paper derives precise estimates for the probability that geodesics in exponential last passage percolation stay within narrow strips, revealing the decay rate of small ball probabilities and deviations, with implications for the directed landscape.

Contribution

It establishes the decay rate of small ball probabilities and small deviations for geodesics in exponential last passage percolation, extending understanding of geodesic fluctuations.

Findings

Probability of geodesic within a narrow strip decays as exp(-Theta(delta^{-3/2})).

Small deviation estimates for the intersection point distribution are linear in delta.

Results are uniform for large n and expected to extend to other models and the directed landscape.

Abstract

For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $\Gamma_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $\Gamma_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $\Gamma_{n}$ and establish that for small $\delta$, the probability that $\Gamma_{n}$ is contained in a strip of width $\delta n^{2/3}$ around the diagonal is $\exp (-\Theta(\delta^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the one point distribution of the geodesic showing that for $\frac{t}{2n}$ bounded away from $0$ and $1$, we have $\mathbb{P}(|x(t)-y(t)|\leq \delta n^{2/3})=\Theta(\delta)$ uniformly in high $n$, where $(x(t),y(t))$ is the unique point where $\Gamma_{n}$ intersects the line $x+y=t$. Our methods are expected to go through for other exactly solvable models of planar last passage percolation and, upon taking the $n\to \infty$ limit, provide analogous estimates for geodesics in the directed landscape.