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

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.