Archive ouverte HAL – Asymptotic behavior of the length of the longest increasing subsequences of random walks

J. Ricardo G. Mendonça 1 Hendrik Schawe 2 Alexander K. Hartmann 3 Alexander Hartmann

J. Ricardo G. Mendonça, Hendrik Schawe, Alexander K. Hartmann, Alexander Hartmann. Asymptotic behavior of the length of the longest increasing subsequences of random walks. Physical Review E , American Physical Society (APS), 2020, 101 (3), ⟨10.1103/PhysRevE.101.032102⟩. ⟨hal-02512208⟩

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 $1/2 \leq \nu \leq 5$. 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 $\theta(\nu \geq 5/2) \approx 0.50$. 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.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. University of Oldenburg
  • 3. Institut für Physik