# Asymptotic behavior of the length of the longest increasing subsequences of random walks

## TL;DR

This paper investigates the asymptotic behavior of the longest increasing subsequence length in random walks with Student's t-distributed steps, revealing a power-law growth with a logarithmic factor and connections to partition theory and Gumbel statistics.

## Contribution

It provides numerical estimates of the asymptotic growth of the LIS length for various step distributions, including heavy-tailed cases, and uncovers links to partition statistics and extreme value distributions.

## Key findings

- Expected LIS length scales as n^θ log n with θ decreasing from 0.70 to 0.50 as ν increases.
- For finite variance steps (ν > 2), confirms E(L_n) ~ √n log n behavior.
- LIS asymptotics resemble those of the largest part in random partitions and follow Gumbel statistics.

## Abstract

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 parameter in the range $1/2 \leq \nu \leq 5$. We find that the expected value $\mathbb{E}(L_{n}) \sim n^{\theta}\ln{n}$ with $\theta$ decreasing from $\theta(\nu=1/2) \approx 0.70$ to $\theta(\nu \geq 5/2) \approx 0.50$. For random walks with distribution of step increments of finite variance ($\nu > 2$), this confirms previous observation of $\mathbb{E}(L_{n}) \sim \sqrt{n}\ln{n}$ 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 $\mathbb{E}(L_{n})$ for random walks with step increments of finite variance.