# Large deviations of the length of the longest increasing subsequence of random permutations and random walks

## TL;DR

This paper uses advanced large-deviation algorithms to analyze the distribution of the longest increasing subsequence in random permutations and walks, confirming known results and exploring new asymptotic behaviors.

## Contribution

It provides numerical verification of the Tracy-Widom distribution for permutations and investigates tail behaviors for random walks where analytical results are lacking.

## Key findings

- Confirmed Tracy-Widom distribution for permutations
- Verified asymptotics of the rate function for permutations
- Observed tail convergence and estimated rate functions for random walks

## Abstract

We study numerically the distributions of the length $L$ of the longest increasing subsequence (LIS) for the two cases of random permutations and of one-dimensional random walks. Using sophisticated large-deviation algorithms, we are able to obtain very large parts of the distribution, especially also covering probabilities smaller than $P(L) = 10^{-1000}$. This enables us to verify for the length of the LIS of random permutations the analytically known asymptotics of the rate function and even the whole Tracy-Widom distribution, to which we observe a rather fast convergence in the larger than typical part. For the length $L$ of LIS of random walks, where no analytical results are known to us, we test a proposed scaling law and observe convergence of the tails into a collapse for increasing system size. Further, we obtain estimates for the leading order behavior of the rate functions of both tails.