Empirical scaling of the length of the longest increasing subsequences of random walks

TL;DR

This paper empirically investigates how the length of the longest increasing subsequences in random walks scales with the number of steps, revealing a range of scaling exponents depending on the step length distribution.

Contribution

It provides Monte Carlo estimates of the scaling behavior of the LIS length for various step distributions, proposing new conjectures and highlighting universality aspects.

Findings

Scaling exponent for heavy-tailed distributions: 0.60 to 0.69.

For finite variance steps, $L_n$ scales as $ ext{sqrt}(n) imes ext{log} n$.

Distribution of $L_n$ follows a universal scaling form for finite variance cases.

Abstract

We provide Monte Carlo estimates of the scaling of the length $L_{n}$ of the longest increasing subsequences of $n$-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, $L_{n} \sim n^{\theta}$ with the leading scaling exponent $0.60 \lesssim \theta \lesssim 0.69$ for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and $\theta \simeq 0.57$ for distributions of finite variance, irrespective of the particular distribution. The results are consistent with existing rigorous bounds for $\theta$, although in a somewhat surprising manner. For random walks with step lengths of finite variance, we conjecture that the correct asymptotic behavior of $L_{n}$ is given by $\sqrt{n}\ln n$, and also propose the form of the subleading asymptotics. The distribution of $L_{n}$ was found to follow a simple scaling form with scaling functions that vary with $\theta$. Accordingly, when the step lengths are of finite variance they seem to be universal. The nature of this scaling remains unclear, since we lack a working model, microscopic or hydrodynamic, for the behavior of the length of the longest increasing subsequences of random walks.