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

For the exactly solvable model of exponential last passage percolation on ${\mathbb{Z}}^2$, consider the geodesic ${\mathrm{\Gamma }}_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of ${\mathrm{\Gamma }}_n$ around the line $x=y$ is $n^{2\mathrm{/}3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for ${\mathrm{\Gamma }}_n$ and establish that for small $\delta$, the probability that ${\mathrm{\Gamma }}_n$ is contained in a strip of width $\delta n^{2\mathrm{/}3}$ around the diagonal is $\exp (-\Theta(\delta^{-3/2}))$uniformlyinhigh$n$. 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 $\mathbb{P}(\mathrm{\mid }x(t)-y(t)\mathrm{\mid }\le \delta n^{2\mathrm{/}3})=\mathrm{\Theta }(\delta )$ uniformly in high $n$, where $(x(t),y(t))$ is the unique point where ${\mathrm{\Gamma }}_n$ 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 $n\to \mathrm{\infty }$ limit, provide analogous estimates for geodesics in the directed landscape.