### Law of Iterated and Fractional Logarithms in Last Passage Percolation

Probability Seminar

22nd July 2024, 3:30 pm – 4:30 pm

Fry Building, G13

Planar exponential last passage percolation (LPP) is one of the very few exactly solvable models for random growth in the Kardar-Parisi-Zhang (KPZ) universality class, for which the weak convergence of the scaled last passage times to the GUE Tracy-Widom distribution is known. Analogous to the classical law of iterated logarithm in case of simple random walk, Ledoux considered the following question: what is the order of growth of the pre-limiting Tracy-Widom fluctuations along the time direction. In this talk we shall present a solution to this problem for both upper and lower deviations using the recently established sharp tail estimates for the largest eigenvalues of $\beta$ ensembles; of which passage time in exponential LPP is a special case owing to a bijection established by Johansson. Our arguments combine these tail estimates with the geometry of exponential LPP landscape and extend the previously established partial results by Ledoux (2016) and Basu et al. (2019). We also consider fluctuations of scaled passage times along the space direction and prove a law of fractional logarithm in that case.

This talk is based on two recent works with Baslingker, Basu, Krishnapur https://doi.org/10.48550/arXiv.2405.12215 and with Basu https://doi.org/10.48550/arXiv.2406.11826.

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