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A new cutoff criterion for non-negatively curved chains

Francesco Pedrotti, Justin Salez

Abstract

The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carré du champ operator, which is specific to differential generators and hence fails on discrete spaces. In the present paper, we show that an approximate version of this chain rule in fact always holds, with an extra cost that depends on the log-Lipschitz regularity of the considered observable. As a consequence, we derive a new cutoff criterion for non-negatively curved chains on finite spaces. The latter allows us to recover, in a simple and unified way, a number of historical instances of cutoff that had been established through model-specific arguments. Emblematic examples include random walk on the hypercube, random transpositions, random walk on the multislice, or MCMC samplers for popular spin systems such as the Ising and Hard-core models on bounded-degree graphs.

A new cutoff criterion for non-negatively curved chains

Abstract

The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carré du champ operator, which is specific to differential generators and hence fails on discrete spaces. In the present paper, we show that an approximate version of this chain rule in fact always holds, with an extra cost that depends on the log-Lipschitz regularity of the considered observable. As a consequence, we derive a new cutoff criterion for non-negatively curved chains on finite spaces. The latter allows us to recover, in a simple and unified way, a number of historical instances of cutoff that had been established through model-specific arguments. Emblematic examples include random walk on the hypercube, random transpositions, random walk on the multislice, or MCMC samplers for popular spin systems such as the Ising and Hard-core models on bounded-degree graphs.
Paper Structure (12 sections, 6 theorems, 66 equations, 1 figure)

This paper contains 12 sections, 6 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The cutoff phenomenon
  3. Bakry-Émery curvature
  4. Main result
  5. Applications
  6. Conjugacy-invariant random walks on groups
  7. Markovian projections
  8. MCMC samplers
  9. Proof
  10. Approximate chain rule
  11. Spatial regularity of the information content
  12. The information-differential route to cutoff

Key Result

Theorem 1

Consider a sequence of Markov triples satisfying (assume:curved)-(assume:symmetry) and for some $\varepsilon\in(0,1)$. Then a cutoff occurs, i.e. for all $\varepsilon\in(0,1)$,

Figures (1)

  • Figure 1: A typical plot of the distance to equilibrium $t\mapsto {\textsc{tv}}(X_t)$. As the ratio $\frac{\mathrm{t}_{\textsc{mix}}(1-\varepsilon)}{\mathrm{t}_{\textsc{mix}}(\varepsilon)}$ approaches $1$, the transition to equilibrium becomes abrupt (cutoff).

Theorems & Definitions (20)

  • Theorem 1: Main result
  • Remark 1: Entropy mixing
  • Example 1: Random walk on the hypercube
  • Example 2: Random transpositions
  • Example 3: Random walks generated by a conjugacy class
  • Lemma 1: Non-negative curvature is preserved under projection
  • Example 4: Ehrenfest model
  • Example 5: Bernoulli-Laplace model
  • Example 6: Random walk on the multislice
  • Example 7: Ising model on a graph
  • ...and 10 more