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Cutoff for mixtures of permuted Markov chains: general case

Bastien Dubail

Abstract

We investigate the mixing properties of a finite Markov chain in random environment defined as a mixture of a deterministic chain and a chain whose state space has been permuted uniformly at random. This work is the counterpart of a companion paper where we focused on a reversible model, which allowed for a few simplifications in the proof. We consider here the general case. Under mild assumptions on the base Markov chains, we prove that with high probability the resulting chain exhibits the cutoff phenomenon at entropic time $\log n / h$, $h$ being some constant related to the entropy of the chain, when the chain is started from a typical state. However contrary to the reversible case uniform cutoff at entropic time does not hold, as we provide an example where the worst-case mixing time has at least polylogarithmic order. We also provide a polylogarithmic upper bound on the worst-case mixing time, which in fact plays a crucial role in deriving the main result for typical states. Incidentally, our proof gives a clear picture of when to expect uniform cutoff at entropic time: it appears as the consequence of a uniform transience property for the covering Markov chain used throughout the proof, which lies on an infinite state space and projects back onto the initial chain.

Cutoff for mixtures of permuted Markov chains: general case

Abstract

We investigate the mixing properties of a finite Markov chain in random environment defined as a mixture of a deterministic chain and a chain whose state space has been permuted uniformly at random. This work is the counterpart of a companion paper where we focused on a reversible model, which allowed for a few simplifications in the proof. We consider here the general case. Under mild assumptions on the base Markov chains, we prove that with high probability the resulting chain exhibits the cutoff phenomenon at entropic time , being some constant related to the entropy of the chain, when the chain is started from a typical state. However contrary to the reversible case uniform cutoff at entropic time does not hold, as we provide an example where the worst-case mixing time has at least polylogarithmic order. We also provide a polylogarithmic upper bound on the worst-case mixing time, which in fact plays a crucial role in deriving the main result for typical states. Incidentally, our proof gives a clear picture of when to expect uniform cutoff at entropic time: it appears as the consequence of a uniform transience property for the covering Markov chain used throughout the proof, which lies on an infinite state space and projects back onto the initial chain.
Paper Structure (62 sections, 37 theorems, 284 equations, 2 figures)

This paper contains 62 sections, 37 theorems, 284 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cutoff for mixtures of permuted Markov Chains
  3. Related works
  4. Proof outline
  5. Dealing with an unknown invariant measure:
  6. The entropic method:
  7. Analysis on the quasi-tree:
  8. Argument for the lower bound:
  9. Concentration of nice trajectories:
  10. Getting a bound on the worst-case mixing time
  11. Organisation of the paper
  12. First moment argument, quasi-trees
  13. Basic notations:
  14. Quenched vs annealed probability, first moment argument
  15. The two-lift chain and half-integer time steps
  16. ...and 47 more sections

Key Result

Theorem 1.1

Let $P_1, P_2$ be two $n \times n$ stochastic matrices, $p_1, p_2 \in M_{n}([0,1])$ be $n \times n$ matrices with entries in $[0,1]$ such that $p_1 + p_2 = 1$ entry-wise, $\sigma$ a uniform permutation of $n$ elements and consider the Markov chain with transition probabilities Suppose Then the chain defined by eq:model0 is irreducible, aperiodic and there exists $h = \Theta(1)$ for which the fol

Figures (2)

  • Figure 1: The finite chains $P_1$, $P_2$ and the quasi-tree $T$ with all centers of type $0$, after identifying each $x \simeq \eta(x)$. The probability to remain at a vertex is not represented.
  • Figure 2: Argument of the proof of Lemma \ref{['lem:mixing_regen']}

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 73 more