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

For the exactly solvable model of exponential last passage percolation on ℤ², consider the geodesic Γ_n joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γ_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 Γ_n and establish that for small δ, the probability that Γ_n is contained in a strip of width δn^2/3 around the diagonal is exp(−Θ(δ^−3/2)) uniformly in high n. We also obtain optimal small deviation estimates for the one point distribution of the geodesic showing that for \({t}\over{2n}\) bounded away from 0 and 1, we have ℙ(∣x(t) − y(t)∣ ≤ δn^2/3) = Θ(δ) uniformly in high n, where (x(t), y(t)) is the unique point where Γ_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 also, upon taking the n → ∞ limit, expected to provide analogous estimates for geodesics in the directed landscape.