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

Mohamed Slim Kammoun

TL;DR

The paper addresses the LCS length of two independent random permutations under conjugation-invariant laws, proving universal $\sqrt{n}$-scale lower bounds and, in the i.i.d. conjugation-invariant setting, asymptotics at least on the $2\sqrt{n}$ scale. The authors develop a proof framework based on Robinson–Schensted correspondence, a cycle-merging Markov operator, and comparisons to uniform and Ewens(0) distributions, enabling precise lower bounds involving the limit-shape functional $\Omega$ and a derived function $G$. They show that with sufficient control of fixed points or cycles, the LCS fluctuations converge to the Tracy–Widom distribution, independent of the second permutation, and they extend results to non-identical laws with explicit lower bounds. These results push toward resolving conjectures about LCS in random permutations, connecting LCS behavior to classical random-Young-diagram limit laws and TW fluctuations, and establishing robust asymptotics under broad invariance assumptions.

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.

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

TL;DR

The paper addresses the LCS length of two independent random permutations under conjugation-invariant laws, proving universal -scale lower bounds and, in the i.i.d. conjugation-invariant setting, asymptotics at least on the scale. The authors develop a proof framework based on Robinson–Schensted correspondence, a cycle-merging Markov operator, and comparisons to uniform and Ewens(0) distributions, enabling precise lower bounds involving the limit-shape functional and a derived function . They show that with sufficient control of fixed points or cycles, the LCS fluctuations converge to the Tracy–Widom distribution, independent of the second permutation, and they extend results to non-identical laws with explicit lower bounds. These results push toward resolving conjectures about LCS in random permutations, connecting LCS behavior to classical random-Young-diagram limit laws and TW fluctuations, and establishing robust asymptotics under broad invariance assumptions.

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 is greater than . We prove in this paper that there exists a universal constant such that their conjecture is satisfied for any pair of i.i.d random permutations of size greater than with distribution invariant under conjugation. We prove also that asymptotically, this expectation is at least of order 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.

Paper Structure

This paper contains 8 sections, 25 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. LCS of two independent random permutations with distribution invariant under conjugation
  3. LCS of two independent random permutations where one of the distributions is invariant under conjugation
  4. Proof of results
  5. General tools
  6. Proof of Proposition \ref{['thm7']} and Corollary \ref{['thmp']}
  7. Proof of Theorem \ref{['thm6']}
  8. Proof of Theorem \ref{['thm8']} and Proposition \ref{['thm5']}.

Key Result

Theorem 1

Assume that for any $n\geq 1$, $\sigma_{1,n}$ and $\sigma_{2,n}$ are independent and their distributions are invariant under conjugation. Then where $\theta$ is the unique solution of $G(2\sqrt{x})=\frac{2+x}{12}$, and

Figures (3)

  • Figure 1: $L_{(7,5,2,1,1,\underline{0})}$
  • Figure 2: Illustration of the Vershik-Kerov-Logan-Shepp convergence
  • Figure 3: $\lambda=(7,2,2,1,1,\underline{0})$ and $\gamma=1$

Theorems & Definitions (43)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • proof
  • Lemma 7
  • Lemma 8
  • proof
  • ...and 33 more