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

TL;DR

This paper numerically studies the longest increasing subsequences of symmetric ultra-fat tailed random walks, revealing a specific scaling behavior and suggesting universality across heavy-tailed distributions.

Contribution

It introduces the numerical analysis of LIS scaling in ultra-fat tailed random walks and demonstrates universality with other heavy-tailed distributions.

Findings

LIS length scales as n^{0.716}

Distribution of LIS length appears universal

Ultra-fat tailed walk behaves like a heavy tailed alpha-stable distribution

Abstract

The longest increasing subsequence (LIS) of a sequence of correlated random variables is a basic quantity with potential applications that has started to receive proper attention only recently. Here we investigate the behavior of the length of the LIS of the so-called symmetric ultra-fat tailed random walk, introduced earlier in an abstract setting in the mathematical literature. After explicit constructing the ultra-fat tailed random walk, we found numerically that the expected length $L_{n}$ of its LIS scales with the length $n$ of the walk like $\langle L_{n} \rangle \sim n^{0.716}$, indicating that, indeed, as far as the behavior of the LIS is concerned the ultra-fat tailed distribution can be thought of as equivalent to a very heavy tailed $\alpha$-stable distribution. We also found that the distribution of $L_{n}$ seems to be universal, in agreement with results obtained for other heavy tailed random walks.