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Cutoff for Rewiring Dynamics on Perfect Matchings

Sam Olesker-Taylor

Abstract

We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs). An $n$-PM is a pairing of $2n$ objects. The $k$-PM RW selects $k$ pairs uniformly at random, disassociates the corresponding $2k$ objects, then chooses a new pairing on these $2k$ objects uniformly at random. The equilibrium distribution is uniform over the set of all $n$-PM. We establish cutoff for the $k$-PM RW whenever $2 \le k \ll n$. If $k \gg 1$, then the mixing time is $\tfrac nk \log n$ to leading order. The case $k = 2$ was established by Diaconis and Holmes (2002) by relating the $2$-PM RW to the random transpositions card shuffle and also by Ceccherini-Silberstein, Scarabotti and Tolli (2007, 2008) using representation theory. We are the first to handle $k > 2$. Our argument builds on previous work of Berestycki, Schramm, Şengül and Zeitouni (2005, 2011, 2019) regarding conjugacy-invariant RWs on the permutation group.

Cutoff for Rewiring Dynamics on Perfect Matchings

Abstract

We establish cutoff for a natural random walk (RW) on the set of perfect matchings (PMs). An -PM is a pairing of objects. The -PM RW selects pairs uniformly at random, disassociates the corresponding objects, then chooses a new pairing on these objects uniformly at random. The equilibrium distribution is uniform over the set of all -PM. We establish cutoff for the -PM RW whenever . If , then the mixing time is to leading order. The case was established by Diaconis and Holmes (2002) by relating the -PM RW to the random transpositions card shuffle and also by Ceccherini-Silberstein, Scarabotti and Tolli (2007, 2008) using representation theory. We are the first to handle . Our argument builds on previous work of Berestycki, Schramm, Şengül and Zeitouni (2005, 2011, 2019) regarding conjugacy-invariant RWs on the permutation group.

Paper Structure

This paper contains 26 sections, 20 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model Set-Up
  3. Mixing and Cutoff Definitions
  4. Main Theorem
  5. Related Previous Work
  6. Acknowledgements
  7. Outline of Approach and Comparison with BS:cutoff-conj-inv
  8. Reductions and Realisations
  9. Cycle Definitions and Reduction to Cycle Structure
  10. From Cycle Structures to Integer Partitions
  11. Decomposing a Perfect Matching into a Sequence of Swaps
  12. Generating a $k$-PM via $k-1$ Swaps
  13. Support Size and Distance from Identity for a Uniform $k$-Rematch
  14. Analysis of Coalescence--Fragmentation Chain
  15. Conditional Uniformity
  16. ...and 11 more sections

Key Result

Theorem 1.5

Let $n \in \mathbb{N}$. Let $k \in \mathbb{N} \setminus \{ 1 \}$ with $k/n\to0$ as $n\to\infty$. Let Then, the $k$-PMRW$M = M_{n, k}$ exhibits cutoff at time $\tt$. In particular, if $k\to\infty$ as $n\to\infty$, then [Officially, this is for a sequence $(k_n)_{n\in\mathbb{N}}$ with $k_n/n \to 0$ as $n\to\infty$ and PMRWs $(M_{n, k_n})_{n\in\mathbb{N}}$.]

Figures (5)

  • Figure 1.1: The rematchings have the following correspondences: \bm{\cdot}\ \text{the 'cross' (left)}\quad \bigl\{ \{ 1,2 \} ,\: \{ 3,4 \} \bigr\} \ \longrightarrow{} \bigl\{ \{ 1,4 \} ,\: \{ 2,3 \} \bigr\} \quad\text{to a}\quad \text{'transposition';}\bm{\cdot}\ \text{the 'bar' (right)}\quad \bigl\{ \{ 1,2 \} ,\: \{ 3,4 \} \bigr\} \ \longrightarrow{} \bigl\{ \{ 1,3 \} ,\: \{ 2,4 \} \bigr\} \quad\text{to a}\quad \text{'reversal'.} \qedhere
  • Figure 3.1: The pair of vertical edges is picked and rematched into either a cross (left) or a bar (right). The bar splits the cycle in two, but the cross does not.
  • Figure 4.1: The first step interacts with only $\{ a, b \}$; we can thus associate $c' = c$. The second step interacts with $\{ b', c' = c \}$; there is no natural way to say whether $b'$ corresponds to $a$ or to $b$. Our algorithm only needs the equality $a \cup b = a' \cup b'$ (as sets): it chooses $U \sim \mathop{\mathrm{Unif}}\nolimits( \{ a',b' \} )$ does a swap with $\{ U, c \}$.
  • Figure 4.2: Four examples from \ref{['exm:decomp:dist-supp']}\ref{['exm:a']}--\ref{['exm:d']}. The dashed lines correspond to the identity matching $\textup{id}\xspace_8$. The solid lines correspond to the matching $\eta$ from the respective examples.
  • Figure 5.1: Two tilings $\lambda$ and $\mu$. The grey shaded tiles are the distinguished tiles, namely $I \in \lambda$ and $J \in \mu$. They have width $| I | = \alpha$ and $| I | = \beta$, respectively. The arrows represent the map $\Phi$ from \ref{['alg:cf:tiling:coupling']}. Four example pairs $(v_i, \Phi(v_i))_{i=1}^4$ are given.

Theorems & Definitions (44)

  • Definition 1.1: Perfect Matching Random Walk
  • Definition 1.2: Total Variation Distance
  • Definition 1.3: Mixing Time
  • Definition 1.4: Cutoff
  • Theorem 1.5: Cutoff for the PM RW
  • Definition 3.1: Cycle Structure
  • Lemma 3.4: TV-Preserving Projection to Cycle Structure
  • Definition 3.5: Coalescence--Fragmentation Chain
  • Lemma 3.6: Evolution of Coalescence--Fragmentation Chain
  • Remark 3.7: Comparison with Random Transpositions
  • ...and 34 more