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\begin{document}$${\widetilde{H}}_N$$\end{document}, N≥1\documentclass[12pt]{minimal}
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\begin{document}$$N \ge 1$$\end{document}, be the point-to-point last passage times of directed percolation on rectangles [(1,1),([γN],N)]\documentclass[12pt]{minimal}
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\begin{document}$$[(1,1), ([\gamma N], N)]$$\end{document} in N×N\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {N}}\times {\mathbb {N}}$$\end{document} over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some αsup>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\sup } >0$$\end{document}, αsup≤lim supN→∞H~N(loglogN)2/3≤(34)2/3\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$\end{document}with probability one, and that αsup=(34)2/3\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}$$\end{document} provided a commonly believed tail bound holds. The result is in contrast with the normalization (logN)2/3\documentclass[12pt]{minimal}
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\begin{document}$$(\log N)^{2/3}$$\end{document} 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 (loglogN)1/3\documentclass[12pt]{minimal}
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\begin{document}$$(\log \log N)^{1/3}$$\end{document} is also discussed.
