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Lower bound for large local transversal fluctuations of Geodesics in Last Passage Percolation (2311.00360v1)
Published 1 Nov 2023 in math.PR
Abstract: For exactly solvable models of planar last passage percolation, it is known that geodesics of length $n$ exhibit transversal fluctuations at scale $n{2/3}$ and matching (up to exponents) upper and lower bounds for the tail probabilities are available. The local transversal fluctuations near the endpoints are expected to be much smaller; it is known that the transversal fluctuation up to distance $r \ll n$ is typically of the order $r{2/3}$ and the probability that the fluctuation is larger than $tr{2/3}$ is at most $Ce{-ct3}$. In this note, we provide a short argument establishing a matching lower bound for this probability.