Non-universality for longest increasing subsequence of a random walk

TL;DR

This paper investigates the behavior of the longest increasing subsequence in symmetric random walks, revealing non-universal exponents that depend on the tail properties of the distribution.

Contribution

It demonstrates that the exponent for the longest increasing subsequence varies for different symmetric random walks, challenging the universality of the $n^{1/2}$ scaling.

Findings

Ultra-fat tailed random walk has a longest increasing subsequence exponent between 0.690 and 0.815.

Symmetric stable-$eta$ distributions with small $eta$ also exhibit exponents greater than 1/2.

The results show non-universality of the LIS scaling in symmetric random walks.

Abstract

The longest increasing subsequence of a random walk with mean zero and finite variance is known to be $n^{1/2 + o(1)}$. We show that this is not universal for symmetric random walks. In particular, the symmetric Ultra-fat tailed random walk has a longest increasing subsequence that is asymptotically at least $n^{0.690}$ and at most $n^{0.815}$. An exponent strictly greater than $1/2$ is also shown for the symmetric stable-$\alpha$ distribution when $\alpha$ is sufficiently small.