A transport approach to the cutoff phenomenon

2025/09/10 by Francesco Pedrotti, Justin Salez, Pedrotti, Francesco +1 · 1 voice · 1 citation
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math.AP #math.PR #math.ST #stat.ML

paper · pdf · doi:10.48550/arxiv.2509.08560

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Substantial progress has recently been made in the understanding of the cutoff phenomenon for Markov processes, using an information-theoretic statistics known as varentropy [Sal23; Sal24; Sal25a; PS25]. In the present paper, we propose an alternative approach which bypasses the use of varentropy and exploits instead a new W-TV transport inequality, combined with a classical parabolic regularization estimate [BGL01; OV01]. While currently restricted to non-negatively curved processes on smooth spaces, our argument no longer requires the chain rule, nor any approximate version thereof. As applications, we recover the main result of [Sal25a] establishing cutoff for the log-concave Langevin dynamics, and extend the conclusion to a widely-used discrete-time sampling algorithm known as the Proximal Sampler.

Citations

• Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon
• Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality
• A new cutoff criterion for non-negatively curved chains
• Cutoff for non-negatively curved diffusions
• Fast Convergence of Φ-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler
• The varentropy criterion is sharp on expanders
• Improved analysis for a proximal algorithm for sampling
• On the Lipschitz Properties of Transportation Along Heat Flows
• Cutoff for non-negatively curved Markov chains
• Structured Logconcave Sampling with a Restricted Gaussian Oracle
• Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry\n Suffices
• The Poincaré inequality and quadratic transportation-variance inequalities
• Log-concavity and strong log-concavity: a review
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Cited by

• Bounded-degree graphs of non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk

Discussions

• A cute little inequality / existence statement from a paper (by somebody else!) last year: arxiv.org/abs/2509.08560 [bsky, 14 points, 2 comments]

Related

• Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature
• A new cutoff criterion for non-negatively curved chains
• The varentropy criterion is sharp on expanders
• Cutoff for non-negatively curved diffusions
• Cutoff for congestion dynamics and related generalized exclusion processes