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

2021/01/05 by Basu, Riddhipratim, Bhatia, Manan
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2101.01717

Abstract

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)/(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, upon taking the n→ ∞ limit, provide analogous estimates for geodesics in the directed landscape.

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