Mixing and cutoff for the systematic scan dynamics of the mean-field ferromagnetic Potts model

2026-07-10

AI Summary

This work addresses the lack of theoretical understanding regarding mixing times for systematic-scan dynamics in mean-field ferromagnetic Potts models. Focusing on the $q$-state model on the complete graph, the authors analyze the systematic-scan Markov chain by leveraging techniques from mixing time analysis, total variation distance estimation, and cutoff phenomenon theory, in conjunction with the model’s phase transition properties. They establish the first general cutoff result for systematic-scan dynamics in spin systems, proving a sharp cutoff at $c(\beta,q)\log n + \Theta(1)$ in the subcritical regime ($\beta < \beta_s$), while demonstrating exponentially slow mixing in the supercritical regime ($\beta > \beta_s$). The bounds are tight in all parameters, thereby filling a significant theoretical gap concerning cutoff behavior for this class of non-reversible global-update chains.

Abstract

We study the mixing time of the systematic scan dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updates a uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics. We take a step toward addressing this imbalance by showing that for every $q\ge 2$ and $β<β_s$, where $β_s$ is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in $Θ(\log n)$ scans or, equivalently, in $Θ(n\log n)$ single site updates. We in fact prove a sharper result; namely, that there exists a constant $c(β,q) > 0$ such that the mixing time is $c(β,q)\log n + Θ(1),$ which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow $Θ(1)$ time window. This result is tight in $β$ as well since the dynamics mixes exponentially slowly for $β> β_s$. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.

Problem

• systematic scan dynamics
• mixing time
• cutoff phenomenon
• Potts model
• mean-field

Innovation

• systematic scan dynamics
• cutoff phenomenon
• mean-field Potts model
• mixing time
• non-reversible Markov chains