Lower Deviations In Beta-Ensembles And Law Of Iterated Logarithm In Last Passage Percolation

For last passage percolation (LPP) on Z(2) 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 ->infinity) T-n-4n/n(1/3)(log log n)(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 beta-Laguerre ensembles, which extends the results proved in the context of the beta-Hermite ensembles by Ledoux and Rider (2010).