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Cutoff Phenomenon for Cyclic Dynamics on Hypercube

Keunwoo Lim

Abstract

The cutoff phenomena for Markovian dynamics have been observed and rigorously verified for a multitude of models, particularly for Glauber-type dynamics on spin systems. However, prior studies have barely considered irreversible chains. In this work, the cutoff phenomenon of certain cyclic dynamics are studied on the hypercube $Σ_{n} = Q^{V_{n}}$, where $Q = \{1, 2, 3\}$ and $V_{n} = \{1,...,n\}$. The main feature of these dynamics is the fact that they are represented by an irreversible Markov chain. Based on the coupling modifications suggested in a previous study of the cutoff phenomenon for the Curie-Weiss-Potts model, a comprehensive proof is presented.

Cutoff Phenomenon for Cyclic Dynamics on Hypercube

Abstract

The cutoff phenomena for Markovian dynamics have been observed and rigorously verified for a multitude of models, particularly for Glauber-type dynamics on spin systems. However, prior studies have barely considered irreversible chains. In this work, the cutoff phenomenon of certain cyclic dynamics are studied on the hypercube , where and . The main feature of these dynamics is the fact that they are represented by an irreversible Markov chain. Based on the coupling modifications suggested in a previous study of the cutoff phenomenon for the Curie-Weiss-Potts model, a comprehensive proof is presented.

Paper Structure

This paper contains 19 sections, 20 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Lower Bound
  4. Preliminaries
  5. Statistical Properties of the Chain
  6. Semi-Synchronized Coupling
  7. Statistics of Stationary Distribution
  8. Proof of Lower Bound
  9. Upper Bound
  10. Coalescing Proportion Chains
  11. Preliminaries
  12. Restriction of Proportion Chain
  13. Semi-Coordinatewise Coupling
  14. Coalescing Basket Chains
  15. Basketwise Coupling
  16. ...and 4 more sections

Key Result

Theorem 1.1

The cyclic dynamics defined on $\Sigma_{n}$ with probability $0<p<1$ exhibit cutoff at mixing time with a window of size $O(n)$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • ...and 12 more