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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 theexponent governing the decay of the small ball probability for $\Gamma_{n}$ andestablish that for small $\delta$, the probability that $\Gamma_{n}$ iscontained 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 smalldeviation estimates for the one point distribution of the geodesic showing thatfor $\frac{t}{2n}$ 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 theline $x+y=t$. Our methods are expected to go through for other exactly solvablemodels of planar last passage percolation and, upon taking the $n\to \infty$limit, provide analogous estimates for geodesics in the directed landscape.

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