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).

有关ℤ最后通道渗滤（LPP）²具有指数通过时间，让Ť _Ñ表示从（1，1）的经过时间到（N，N-）。我们调查了序列{迭代的对数律牛逼_ñ } _{ñ ≥1} ; 我们显示\ {\ lim \，{\ inf _ {n \ to \ infty}} {{{T_n}-4n} \ over {{n ^ {1/3}} {{\ left（{\ log \ ，\ log \，n} \ right）} ^ {1/3}}}} \）几乎可以肯定地收敛到确定性负常数，并获得相同的一些估计。这解决了Ledoux（2018）的一个猜想，其中证明了相关的下界和相应的上尾巴的相似结果。我们的证明依赖于从点到点通过时间到考虑点到线通过时间的角度上的微小转变，并利用后者与拉盖尔正交合奏（LOE）的最大特征值的对应关系。具有独立利益的关键技术要素是β- Laguerre集合最大特征值的较低尾巴偏离概率的新下界，这扩展了Ledoux和Rider在β-赫尔曼特集合中所证明的结果（ 2010）。