A Law of the Iterated Logarithm for Directed Last Passage Percolation

Abstract

Let ${\widetilde{H}}_N$, $N \ge 1$, be the point-to-point last passage times of directed percolation on rectangles $[(1,1), ([\gamma N], N)]$ in ${\mathbb {N}}\times {\mathbb {N}}$ over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some $\alpha _{\sup } >0$,

$$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$

with probability one, and that $\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}$ provided a commonly believed tail bound holds. The result is in contrast with the normalization $(\log N)^{2/3}$ for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed $(\log \log N)^{1/3}$ is also discussed.