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Cutoff for mixtures of permuted Markov chains: reversible case

Bastien Dubail

Abstract

We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been permuted uniformly at random. It generalizes in particular a result of Hermon, Sly and Sousi, who proved the cutoff phenomenon at entropic time for the simple random walk on a graph with an added uniform matching. 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. We note that the results presented here are the consequence of a work conducted for a more general model that does not assume reversibility, which will be the object of a companion paper. Thus, most of our proofs do not actually require reversibility, which constitutes an important technical contribution. Finally, our argument relies on a novel concentration result for "low-degree" functions on the symmetric group, established specifically for our purpose but which could be of independent interest.

Cutoff for mixtures of permuted Markov chains: reversible case

Abstract

We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been permuted uniformly at random. It generalizes in particular a result of Hermon, Sly and Sousi, who proved the cutoff phenomenon at entropic time for the simple random walk on a graph with an added uniform matching. 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. We note that the results presented here are the consequence of a work conducted for a more general model that does not assume reversibility, which will be the object of a companion paper. Thus, most of our proofs do not actually require reversibility, which constitutes an important technical contribution. Finally, our argument relies on a novel concentration result for "low-degree" functions on the symmetric group, established specifically for our purpose but which could be of independent interest.
Paper Structure (61 sections, 39 theorems, 283 equations, 1 figure)

This paper contains 61 sections, 39 theorems, 283 equations, 1 figure.

Table of Contents

  1. Models and main results
  2. Cutoff for mixtures of reversible permuted Markov Chains
  3. About reversibility
  4. A concentration inequality for low-degree functions on the symmetric group
  5. Related works
  6. Proof outline
  7. Knowledge of the invariant measure:
  8. The entropic method:
  9. Analysis on the quasi-tree:
  10. Argument for the lower bound:
  11. Concentration of nice trajectories:
  12. Computation of $\hat{\pi}$:
  13. Organisation of the paper
  14. First moment argument, quasi-trees
  15. Basic notations:
  16. ...and 46 more sections

Key Result

Theorem 1.1

Let $n \geq 1$ be an integer, $\sigma$ a permutation of $n$ elements chosen uniformly at random, $(G_1,c_1), (G_2,c_2)$ two electrical networks with common vertex set $[n]$ and $\alpha, \beta > 0$. Then consider the Markov chain on $[n]$ defined by the electrical network $(G^{\ast}, c)$ with conduct Suppose Then there exists $h=h(n)$ bounded from above and away from $0$ for which the following ho

Figures (1)

  • Figure 1: Argument of the proof of Lemma \ref{['lem:mixing_regen']}

Theorems & Definitions (89)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.1
  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • ...and 79 more