Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Information-theoretic classification of the cutoff phenomenon in Markov processes

Youjia Wang, Michael C. H. Choi

TL;DR

This work provides a unified, information-theoretic framework for the cutoff phenomenon in Markov processes by classifying divergences into four types and proving that cutoff behavior is equivalent within each type. It introduces the $\mathcal{F}_{p,q}$ family to connect $f$-divergences with Renyi divergences and establishes product-condition criteria that guarantee cutoff, extending results to non-reversible and non-normal settings via a functional-analytic approach. The paper also develops practical spectral-gap-based criteria, including $\pi$-weighted divergences and nonlinear constants, and demonstrates sharp separations between divergence types through classic counterexamples. Together, these results provide a comprehensive toolkit to analyze and compare cutoff phenomena across a wide range of probability metrics. The findings have broad implications for understanding convergence to equilibrium in complex stochastic systems and for guiding the choice of divergence in cutoff analysis.

Abstract

We investigate the cutoff phenomenon for Markov processes under information divergences such as $f$-divergences and Rényi divergences. We classify most common divergences into four types, namely $L^2$-type, $\mathrm{TV}$-type, separation-type and $\mathrm{KL}$ divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.

Information-theoretic classification of the cutoff phenomenon in Markov processes

TL;DR

This work provides a unified, information-theoretic framework for the cutoff phenomenon in Markov processes by classifying divergences into four types and proving that cutoff behavior is equivalent within each type. It introduces the family to connect -divergences with Renyi divergences and establishes product-condition criteria that guarantee cutoff, extending results to non-reversible and non-normal settings via a functional-analytic approach. The paper also develops practical spectral-gap-based criteria, including -weighted divergences and nonlinear constants, and demonstrates sharp separations between divergence types through classic counterexamples. Together, these results provide a comprehensive toolkit to analyze and compare cutoff phenomena across a wide range of probability metrics. The findings have broad implications for understanding convergence to equilibrium in complex stochastic systems and for guiding the choice of divergence in cutoff analysis.

Abstract

We investigate the cutoff phenomenon for Markov processes under information divergences such as -divergences and Rényi divergences. We classify most common divergences into four types, namely -type, -type, separation-type and divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.
Paper Structure (16 sections, 20 theorems, 177 equations, 1 table)

This paper contains 16 sections, 20 theorems, 177 equations, 1 table.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Preliminaries
  4. Markov process
  5. Cutoff phenomenon and $f$-divergence
  6. $L^p$-cutoff
  7. Reversible cases
  8. $\mathcal{F}_{p,q}$ family and Rényi divergence with $\alpha\in [2,\infty]$
  9. $\alpha$-divergence and Rényi divergence with $1<\alpha\leq 2$
  10. KL divergence and total variation distance
  11. $\mathrm{TV}$-type $f$-divergences and Rényi divergence with $0<\alpha<1$
  12. Separation-type divergences
  13. Examples and counter examples
  14. Non-reversible cases
  15. Normal cases
  16. ...and 1 more sections

Key Result

Proposition 2.1

Assume a Markov process have a semigroup $P_t$, stationary distribution $\pi$ and spectral gap $\lambda\geq 0$. For all $f\in L^2(\mathcal{X},\pi)$ and $t\in T$, we have where $\kappa$ is the second largest singular value of $P_1$. Moreover, if $P_t: L^2(\mathcal{X},\pi)\rightarrow L^2(\mathcal{X},\pi)$ is normal, we have where $\Pi f(x):=\pi(f), \forall x\in \mathcal{X}$.

Theorems & Definitions (51)

  • Definition 2.1: Spectral gap of Markov process
  • Proposition 2.1: Convergence rate of Markov semigroup
  • Definition 2.2: Cutoff phenomenon, chen2008cutoff
  • Proposition 2.2: Cutoff and mixing time, chen2008cutoff, Proposition 2.3
  • Definition 2.3: Csiszár's $f$-divergence
  • Proposition 2.3: Some properties of information divergences
  • Proposition 2.4: Some properties of $L^p(\mathcal{X},\pi)$ and $d_p(x,t)$, chen2008cutoff, dunford1988linear
  • Proposition 2.5: Riesz-Thorin Interpolation Theorem, stein2011functional, bernard2013interpolation
  • Proposition 2.6: Characterization of $L^p$-cutoff, chen2008cutoff, Theorem 5.3, 5.4
  • Definition 3.1: $\mathcal{F}_{p,q}$ family
  • ...and 41 more