2019/09/03 by Riddhipratim Basu, Shirshendu Ganguly, Basu, Riddhipratim +5
Computer Science · Mathematics · Psychology · #Bayesian Methods and Mixture Models #FOS: Mathematics #Iterated function #Iterated logarithm #Law of the iterated logarithm #Logarithm #Mathematical analysis #Mathematics #Percolation (cognitive psychology) #Percolation threshold #Physics #Probability (math.PR) #Psychology #Quantum mechanics #Random Matrices and Applications #Statistical physics #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1909.01333
21 pages, 1 figure
arxiv created 2019/09/03 · openalex publication_date 2019/09/03 · arxiv updated 2019/09/04 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
For the last passage percolation (LPP) on ℤ2 with exponential passage times, let Tn denote the passage time from (1,1) to (n,n). We investigate the law of iterated logarithm of the sequence \Tn\n≥ 1; we show that \liminfn→ ∞ \fracTn-4nn1/3(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 (J. Theor. Probab., 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 (Electron. J. Probab., 2010).