Lower deviations in β-ensembles and law of iterated logarithm in last passage percolation

Abstract

For last passage percolation (LPP) on ℤ² with exponential passage times, let T_n denote the passage time from (1, 1) to (n,n). We investigate the law of iterated logarithm of the sequence {T_n}_n≥1; we show that $\lim \,{\inf _{n \to \infty }}{{{T_n} - 4n} \over {{n^{1/3}}{{\left( {\log \,\log \,n} \right)}^{1/3}}}}$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight shift in perspective from point-to-point passage times to considering point-to-line passage times instead, and exploiting the correspondence of the latter to the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient, which is of independent interest, is a new lower bound of lower tail deviation probability of the largest eigenvalue of β-Laguerre ensembles, which extends the results proved in the context of the β-Hermite ensembles by Ledoux and Rider (2010).