# On the longest common subsequence of independent random permutations invariant under conjugation

## TL;DR

This paper proves that for large permutations invariant under conjugation, the expected length of their longest common subsequence grows at least as fast as 2√n, confirming a conjecture and analyzing fluctuations.

## Contribution

It establishes a universal lower bound for the LCS of conjugation-invariant permutations and characterizes its asymptotic fluctuations as Tracy-Widom type.

## Key findings

- Expected LCS is at least 2√n asymptotically.
- Confirms Bukh and Zhou's conjecture for large conjugation-invariant permutations.
- Fluctuations of LCS follow Tracy-Widom distribution under certain conditions.

## Abstract

Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$ such that their conjecture is satisfied for any pair of i.i.d random permutations of size greater than $n_1$ with distribution invariant under conjugation. We prove also that asymptotically, this expectation is at least of order $2\sqrt{n}$ which is the asymptotic behaviour of the uniform setting. More generally, in the case where the laws of the two permutations are not necessarily the same, we gibe a lower bound for the expectation. In particular, we prove that if one of the permutations is invariant under conjugation and with a good control of the expectation of the number of its cycles, the limiting fluctuations of the length of the longest common subsequence are of Tracy-Widom type. This result holds independently of the law of the second permutation.