A Law of the Iterated Logarithm for Directed Last Passage Percolation
A Law of the Iterated Logarithm for Directed Last Passage Percolation
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DOI：
10.1007/s10959-017-0775-z
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发表时间：
2016-10
影响因子：
0.8
通讯作者：
M. Ledoux
中科院分区：
文献类型：
--
作者：
M. Ledoux
Let,, be the point-to-point last passage times of directed percolation on rectanglesinover exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some, $$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$with probability one, and thatprovided a commonly believed tail bound holds. The result is in contrast with the normalizationfor 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 speedis also discussed.
Let,, be the point-to-point last passage times of directed percolation on rectanglesinover exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some, $$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$with probability one, and thatprovided a commonly believed tail bound holds. The result is in contrast with the normalizationfor 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 speedis also discussed.