Abstract
For the exactly solvable model of exponential last passage percolation on ℤ2, consider the geodesic Γn joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γn around the line x = y is n2/3+o(1) with high probability. We obtain the exponent governing the decay of the small ball probability for Γn and establish that for small δ, the probability that Γn is contained in a strip of width δn2/3 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)∣ ≤ δn2/3) = Θ(δ) uniformly in high n, where (x(t), y(t)) is the unique point where Γn 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.
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Acknowledgements
The authors thank an anonymous referee for a careful review of the manuscript and useful comments and Sudeshna Bhattacharjee for helpful comments on a previous version of the paper. R. B. is partially supported by a Ramanujan Fellowship (SB/S2/RJN-097/2017) and a MATRICS grant (MTR/2021/000093) from the Science and Engineering Research Board, an ICTS-Simons Junior Faculty Fellowship, DAE project no. RTI4001 via ICTS, and the Infosys Foundation via the Infosys-Chandrasekharan Virtual Centre for Random Geometry of TIFR. M. B. is supported by a Peterson Fellowship at MIT. The first version of this work was completed while MB was a student at the Long Term Visiting Students Program (LTVSP) at ICTS. He gratefully acknowledges the support.
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Basu, R., Bhatia, M. Small deviation estimates and small ball probabilities for geodesics in last passage percolation. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2635-8
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DOI: https://doi.org/10.1007/s11856-024-2635-8