Asymptotic behavior of the length of the longest increasing subsequences of random walks · arXivDesk
1907.00486Jun 30, 20198 pages (REVTeX 4.1, twocolumn), several figures, 29 references, some conjectural thoughts. This version identical to the published one (minor differences are intentional)
Asymptotic behavior of the length of the longest increasing subsequences of random walks
J. Ricardo G. Mendonça, Hendrik Schawe, Alexander K. Hartmann
We numerically estimate the leading asymptotic behavior of the length Ln of the longest increasing subsequence of random walks with step increments following Student's t-distribution with parameter in the range 1/2≤ν≤5. We find that the expected value E(Ln)∼nθlnn
Nearby in the stack
with
θ
decreasing from
θ(ν=1/2)≈0.70
to
θ(ν≥5/2)≈0.50
. For random walks with distribution of step increments of finite variance (
ν>2
), this confirms previous observation of
E(Ln)∼nlnn
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(Ln)
for random walks with step increments of finite variance.