Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon (2508.21055v1)

Abstract: The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of these lecture notes is to provide a self-contained introduction to this fascinating question, and to describe its recently-uncovered relations with entropy, curvature and concentration.

• The paper demonstrates that incorporating varentropy and curvature yields a universal condition for cutoff in non-negatively curved chains.
• It applies operator theory and functional inequalities (Poincaré, Log-Sobolev) to derive explicit mixing time bounds for canonical models like the n-cube.
• The study bridges geometric, spectral, and information-theoretic methods, unifying cutoff analysis across high-dimensional stochastic processes.

Modern Aspects of Markov Chains: Entropy, Curvature, and the Cutoff Phenomenon

Introduction

This paper provides a comprehensive and rigorous treatment of the cutoff phenomenon in finite Markov chains, emphasizing its connections to entropy, curvature (Ollivier-Ricci and Bakry-Émery), and functional inequalities. The analysis is grounded in operator theory, information theory, and discrete geometry, and the exposition is tailored to researchers interested in the quantitative analysis of mixing times, phase transitions, and concentration phenomena in high-dimensional stochastic processes.

The Cutoff Phenomenon: Formalization and Examples

The cutoff phenomenon is defined as an abrupt transition in the convergence to equilibrium for certain Markov chains as the state space size diverges. Rather than a gradual decay, the total variation distance to equilibrium remains near maximal for a significant period and then drops sharply at a critical threshold. The paper formalizes this using the worst-case total variation distance $(t)$ and the mixing time $t_{mix}(\varepsilon)$, and provides explicit asymptotics for two canonical models:

• Random Walk on the $n$-Cycle: Mixing occurs smoothly on the $n^2$ timescale, with the limiting profile $F(t)$ given by an integral over the local CLT density.

  Figure 1: The smooth limiting profile $F$ for simple random walk on the $n$-cycle.

• Random Walk on the $n$-Cube: Exhibits a sharp cutoff at $t = (n \log n)/4$, with the transition window width $O(n)$ and the profile governed by the error function.

The paper highlights that cutoff is not universal: while the $n$-cube shows cutoff, the $n$-cycle does not, and the product condition (Peres' criterion) is necessary but not sufficient for cutoff. Rank-one perturbations and certain sparse expanders provide explicit counterexamples.

Functional Inequalities and Mixing Time Bounds

The analysis leverages functional inequalities to bound mixing times:

• Poincaré Inequality: Relates the Dirichlet form to variance, with the spectral gap $\gamma$ controlling the exponential decay of variance and providing mixing time bounds.
• Modified Log-Sobolev Inequality (MLSI): Controls entropy decay, with the constant $\alpha$ yielding sharper mixing time estimates, especially in high-dimensional product spaces.
• Log-Sobolev Inequality (LSI): Equivalent to hypercontractivity for reversible chains, with the constant $\beta$ related to $\alpha$ and $\gamma$ via $4\beta \leq \alpha \leq 2\gamma$.

The paper provides explicit spectral decompositions for reversible chains and demonstrates the hierarchy and near-saturation of these constants in the $n$-cube and $n$-cycle models.

Curvature: Ollivier-Ricci and Bakry-Émery

The geometric perspective is developed via discrete analogues of Ricci curvature:

• Ollivier-Ricci Curvature ($\kappa$): Defined via contraction of Wasserstein distances between transition kernels. Positive curvature implies exponential decay of Lipschitz norms and provides mixing time bounds of the form $t_{mix} \leq \kappa^{-1} \log(diam(X)/\varepsilon)$.
• Bakry-Émery Curvature ($\rho$): Characterized by the $\mathrm{CD}(\rho,\infty)$ criterion, relating the iterated carré du champ operator to the Dirichlet form. Positive curvature yields local Poincaré inequalities and sub-commutation of quadratic forms.

The paper establishes that for random walks on Abelian groups and conjugacy-invariant walks, both curvatures are non-negative, and the spectral gap is bounded below by curvature (Lichnerowicz-type estimates).

Information-Theoretic Approach: Varentropy and the Mixing Window

A novel contribution is the use of varentropy (variance of information content) to control the width of the mixing window, providing a second-order criterion for cutoff. The reversed Pinsker inequality is used to relate entropy and total variation, and the paper proves that for chains with non-negative curvature, the mixing window width is bounded in terms of varentropy and the spectral gap:

$$w_{mix}(\varepsilon) \leq \frac{2}{\gamma \varepsilon^2} (1 + \sqrt{V_\varepsilon})$$

where $V_\varepsilon$ is the worst-case varentropy at the pre-cutoff time. This leads to a universal sufficient condition for cutoff: if $$\gamma w_{mix}(\varepsilon) \gg 1 + \sqrt{V_\varepsilon}$$, cutoff occurs.

Sufficient Criteria for Cutoff in Non-Negatively Curved Chains

The paper proves that for weakly reversible chains with non-negative Bakry-Émery curvature, cutoff occurs as soon as the spectral gap satisfies $\gamma \gg \log d$ (where $d$ is the sparsity parameter), or the MLSI constant satisfies $\alpha \gg \log \log d$. This criterion is shown to be satisfied in a wide range of models, including:

• Random walks on the $n$-cube and random transpositions
• Conjugacy-invariant walks with bounded complexity
• High-temperature Ising and hard-core models on bounded-degree graphs

  Figure 2: The blue vertices form an independent set with the largest possible size.

Implications and Open Problems

The results unify and generalize previous model-specific proofs of cutoff, providing a framework that connects geometric, spectral, and information-theoretic properties. The varentropy approach and curvature-based criteria offer practical tools for predicting cutoff in high-dimensional, rapidly mixing systems, especially those with product-like or weakly dependent structure.

Open problems remain, notably the extension of these criteria to negatively curved chains (e.g., random walks on expanders), the characterization of cutoff in vertex-transitive graphs, and the development of model-independent necessary and sufficient conditions for cutoff.

Conclusion

This work synthesizes modern advances in the theory of Markov chain mixing, demonstrating deep connections between entropy, curvature, and the cutoff phenomenon. The varentropy-based criterion and curvature estimates provide robust, quantitative tools for analyzing phase transitions in mixing, with broad applicability to random walks, interacting particle systems, and MCMC algorithms. The theoretical framework established here sets the stage for further progress on universality and sharp thresholds in high-dimensional stochastic dynamics.