Mendonça J Ricardo G, Schawe Hendrik, Hartmann Alexander K
Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, 03828-000 São Paulo, Brazil.
LPTMS, CNRS UMR 8626, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay CEDEX, France.
Phys Rev E. 2020 Mar;101(3-1):032102. doi: 10.1103/PhysRevE.101.032102.
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 parameters in the range 1/2≤ν≤5. We find that the expected value E(L_{n})∼n^{θ}lnn, with θ decreasing from θ(ν=1/2)≈0.70 to θ(ν≥5/2)≈0.50. For random walks with a distribution of step increments of finite variance (ν>2), this confirms previous observation of E(L_{n})∼sqrt[n]lnn 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 E(L_{n}) for random walks with step increments of finite variance.
