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About universality of large deviation principles for conjugacy invariant permutations

Alice Guionnet, Mohamed Slim Kammoun

TL;DR

We study universality of large deviation principles for conjugacy-invariant random permutations with a sharp control on cycle numbers. By introducing the CI_{\alpha,\beta} class and developing an exponential-approximation framework via a one-step Markov coupling to Ewens(0), we show that LDPs proven for the uniform measure extend to a broad class of CI distributions, including Ewens measures. This yields universal LDPs for statistics such as the longest increasing subsequence (LIS) with speeds $\sqrt{n}$ and $n$ and explicit rate functions $I_{\mathrm{LIS},\frac12}$ and $I_{\mathrm{LIS},1}$, as well as for Eulerian statistics and joint descent-type statistics; we also address edge behavior in the RSK framework. The results reveal that cycle structure, rather than the full microscopic detail of the permutation, governs the rare-event behavior across a wide universality class, offering a unified perspective with potential links to random-matrix phenomena.

Abstract

We prove the universality of the large deviations for conjugacy invariant permutations with few cycles. As an application, we establish the universality of large deviation at speeds $n$ and $\sqrt{n}$ for the length of monotone subsequences in conjugacy invariant permutations, with a sharp control over the total number of cycles. This universality class includes the well-known Ewens measures.

About universality of large deviation principles for conjugacy invariant permutations

TL;DR

We study universality of large deviation principles for conjugacy-invariant random permutations with a sharp control on cycle numbers. By introducing the CI_{\alpha,\beta} class and developing an exponential-approximation framework via a one-step Markov coupling to Ewens(0), we show that LDPs proven for the uniform measure extend to a broad class of CI distributions, including Ewens measures. This yields universal LDPs for statistics such as the longest increasing subsequence (LIS) with speeds and and explicit rate functions and , as well as for Eulerian statistics and joint descent-type statistics; we also address edge behavior in the RSK framework. The results reveal that cycle structure, rather than the full microscopic detail of the permutation, governs the rare-event behavior across a wide universality class, offering a unified perspective with potential links to random-matrix phenomena.

Abstract

We prove the universality of the large deviations for conjugacy invariant permutations with few cycles. As an application, we establish the universality of large deviation at speeds and for the length of monotone subsequences in conjugacy invariant permutations, with a sharp control over the total number of cycles. This universality class includes the well-known Ewens measures.
Paper Structure (17 sections, 18 theorems, 79 equations)

This paper contains 17 sections, 18 theorems, 79 equations.

Table of Contents

  1. Introduction and main results
  2. Monotone subsequences
  3. Strategy of the proofs
  4. Applications
  5. Applications to the longest increasing subsequence.
  6. Applications to large deviations for the Eulerian statistics
  7. Applications to the joint large deviations for the descents and the inverse descents
  8. Edge of RSK
  9. Some comments
  10. Proof of the results
  11. Proof of Property \ref{['RQ:controle_functions']}
  12. Proof of Property $(\ref{['eq:cycle_controle']})$ for Ewens distributions
  13. Proof of Theorem \ref{['Prob:Meta theorem']}
  14. Proof of Theorem \ref{['Main_theorem_LIS']}
  15. Proof of Proposition \ref{['Prop: Ewens_LIS']}
  16. ...and 2 more sections

Key Result

Theorem 3

MR1653841deuschel_zeitouni_1999 For any $x \ge 2$, and for any $0<x<2$, where

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Lemma 7
  • Corollary 9
  • proof
  • Corollary 10
  • ...and 20 more