A Law of the Iterated Logarithm for Directed Last Passage Percolation

Let $${\widetilde{H}}_N$$ H ~ N , $$N \ge 1$$ N ≥ 1 , be the point-to-point last passage times of directed percolation on rectangles $$[(1,1), ([\gamma N], N)]$$ [ ( 1 , 1 ) , ( [ γ N ] , N ) ] in $${\mathbb {N}}\times {\mathbb {N}}$$ N × N over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some $$\alpha _{\sup } >0$$ α sup > 0 , $$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$ α sup ≤ lim sup N → ∞ H ~ N ( log log N ) 2 / 3 ≤ ( 3 4 ) 2 / 3 with probability one, and that $$\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}$$ α sup = ( 3 4 ) 2 / 3 provided a commonly believed tail bound holds. The result is in contrast with the normalization $$(\log N)^{2/3}$$ ( 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 $$(\log \log N)^{1/3}$$ ( log log N ) 1 / 3 is also discussed.