Asymptotic behavior of the length of the longest increasing subsequences of random walks

Abstract

We numerically estimate the leading asymptotic behavior of the length $L_n$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2\le \nu \le 5$. We find that the expected value $\mathbb{E}(L_n)\sim n^{\theta }\ln n$ with $\theta$ decreasing from $\theta (\nu =1/2)\approx 0.70$ to $\theta (\nu \ge 5/2)\approx 0.50$. For random walks with distribution of step increments of finite variance ($\nu >2$), this confirms previous observation of $\mathbb{E}(L_n)\sim \sqrt{n}\ln n$ to leading order. We note that this asymptotic behavior (including the subleading term) resembles that of the largest part of random integer partitions under the uniform measure and that, curiously, both random variables seem to follow Gumbel statistics. We also provide more refined estimates for the asymptotic behavior of $\mathbb{E}(L_n)$ for random walks with step increments of finite variance.

8 pages (REVTeX 4.1, twocolumn), several figures, 29 references, some conjectural thoughts. This version identical to the published one (minor differences are intentional)

References in corpus (5)

• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Level Density of a Bose Gas and Extreme Value Statistics
• Integer Partitions and Exclusion Statistics
• Large deviations of the length of the longest increasing subsequence of random permutations and random walks
• Empirical scaling of the length of the longest increasing subsequences of random walks

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• A numerical investigation into the scaling behavior of the longest increasing subsequences of the symmetric ultra-fat tailed random walk