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

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.

• The paper establishes sharp cutoff in systematic scan dynamics with explicit mixing time constants for the mean-field ferromagnetic Potts model.
• It employs a three-phase analysis—burn-in, contractive coupling, and final coalescence—using martingale decompositions and precise spectral estimates.
• The findings demonstrate at least a twofold speedup over Glauber dynamics, validating deterministic updates in high-throughput MCMC simulations.

Mixing and Cutoff for Systematic Scan Dynamics in the Mean-Field Ferromagnetic Potts Model

Introduction and Context

This work provides a comprehensive analysis of the mixing time and cutoff phenomenon for the systematic scan (deterministic sweep) single-site heat-bath dynamics in the mean-field ferromagnetic $q$-state Potts model. The model considers the $n$-vertex complete graph with interaction parameter $\beta < \beta_s(q)$ (the spinodal or metastability threshold). Unlike the well-studied Glauber (random scan) dynamics, the systematic scan is both non-reversible and global, with updates deterministically sweeping through the vertex set. While systematic scan is empirically favored due to parallelism and memory locality, theoretical results on its convergence remain substantially underspecified compared to the random scan, particularly concerning sharp mixing time asymptotics and cutoff behavior.

Main Results

The central result establishes that, for any $q \geq 2$ and any $\beta < \beta_s$, the systematic scan dynamics exhibits cutoff at mixing time $T_{\text{mix}} = c(\beta, q) \log n + \Theta(1)$, where $c(\beta, q)$ is explicitly characterized by the equation

$\beta (1 - e^{-b})/(qb) = e^{-b},$

with $c(\beta,q) = 1/(2b(\beta,q))$. The window of cutoff is $O(1)$ in the number of full scans (i.e., $n$0 in single-site updates), matching the qualitative behavior observed for the Glauber dynamics but with at least a twofold speedup in leading constant.

When $n$1, it is proved that the dynamics slows exponentially as with Glauber, and therefore the mixing time bound is essentially tight in all parameters.

This result is, to the best of current knowledge, the first sharp cutoff analysis for (non-randomized) systematic scan dynamics in any standard mean-field spin system.

Technical Contributions

The analysis consists of three main phases:

1. Burn-in Phase: After $n$2 full scans, the configuration's proportion vector becomes close (in $n$3 and finer senses) to the uniform vector $n$4, with high probability. This proximity is quantified via a set $n$5 capturing fine-grained "spread" properties over ordered sections, essential given non-reversibility and the deterministic update order. A detailed martingale decomposition and tailored maximal Azuma inequality control fluctuations and show concentration.
2. Contractive Coupling Phase: Coupling two chains in $n$6 and applying an optimal per-step site-coupling, the Hamming distance and the $n$7 distance to $n$8 are shown to contract geometrically at a rate precisely determined by the spectral parameter $n$9. The coupling rate analysis is meticulous, relying on recursive analysis of the evolution of the occupation vectors and tight Taylor approximations—yielding the explicit leading constant in the mixing time.
3. Final Coalescence: When the chains are within $\beta < \beta_s(q)$0 Hamming distance, Pinsker's inequality and precise control over the KL divergence between the chains’ distributions (after one systematic scan) ensure that full coalescence occurs with high probability in one additional scan.

Both the upper and lower bounds are tight up to the $\beta < \beta_s(q)$1-window, and the method provides a full proof of the cutoff phenomenon.

Numerical and Asymptotic Aspects

The explicit calculation of $\beta < \beta_s(q)$2 confirms a significant speedup relative to random-scan (Glauber) dynamics, consistent with folklore conjecture and practical observations. Notably, for $\beta < \beta_s(q)$3 and $\beta < \beta_s(q)$4, the leading constants are separated by at least a factor of 2. This quantifies, for the mean-field setting, the maximal efficiency gain through systematic rather than random scan orderings.

Comparison to Prior Work

Existing results for the mean-field Potts model have almost exclusively focused on Glauber dynamics; see, e.g., the sharp cutoff at $\beta < \beta_s(q)$5 for random scan [Cuff et al., J. Stat. Phys., 2012]. For systematic scan, previous analyses provided only coarse $\beta < \beta_s(q)$6 upper bounds for the sub-critical regime and no sharp cutoff results. Despite an emerging literature on systematic versus random scan mixing time comparisons and spectral gap “solidarity,” none previously yielded sharp asymptotics or cutoff proofs for non-reversible, global update dynamics in high-dimensional models.

Theoretical and Practical Implications

This work resolves a key open problem in the mixing analysis of MCMC algorithms for mean-field spin systems under deterministic update rules. The establishment of cutoff for a global, non-reversible chain exposes a regime where non-randomization, often seen as analytically obstructive, can yield both optimal empirical and provable efficiency. The tools developed—martingale-based fluctuation bounds, multiphase coupling arguments, and precise recursive analysis—should be extensible to other high-symmetry, high-entropy settings and possibly to other forms of deterministic or block-update dynamics.

From a Markov chain theory viewpoint, the demonstration of cutoff in a deterministic, non-reversible high-dimensional setting advances understanding of the universality and limits of the cutoff phenomenon beyond classical random-scan, reversible scenarios.

On the practical side, these results rigorously support the choice of systematic scan in high-throughput simulation contexts, justifying both empirical speedups and the selectivity of update scheduling in parallel inference methods.

Future Directions

Future directions include extension to other non-reversible MCMC schemes (e.g., broad classes of scan orderings beyond the fixed sequential one, randomized block updates), analysis for non-mean-field graphs including high-degree expanders, and potential applications of these contraction estimates in bounding mixing of systematic scan for non-monotone or constrained combinatorial models.

Additional research can aim to close the gap at criticality ($\beta < \beta_s(q)$7), where it is conjectured that mixing slows to a nontrivial polynomial rate and the cutoff ceases, in line with phase transition theory.

Conclusion

This paper establishes optimal mixing and identifies cutoff with explicit location and window for the systematic scan dynamics in the mean-field ferromagnetic Potts model for all regimes where rapid mixing holds. This result corrects a prominent imbalance in understanding between Glauber and deterministic-scan dynamics in spin systems and introduces robust techniques that can drive further progress in non-reversible and MCMC theory for high-dimensional statistical models.