Optimal mixing of the systematic scan dynamics via approximate tensorization of entropy

Abstract

We study the mixing time of the systematic scan dynamics for high-dimensional discrete distributions. This Markov chain updates coordinates sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics that updates coordinates selected uniformly at random. The systematic scan is often favored in practice because it exhibits strong empirical performance, but its theoretical analysis remains far less developed than that of Glauber dynamics. We take a step toward addressing this imbalance by showing that two standard functional notions of weak dependence between the coordinates of the distribution provide strong convergence guarantees for the systematic scan dynamics. First, we show that approximate tensorization of entropy implies optimal $O(\log n)$ mixing time for every scan order under standard marginal, connectivity, and bounded interaction degree assumptions about the distribution. Second, we show that approximate tensorization of variance yields a constant-factor contraction of the variance functional per scan, which in turn implies an optimal $O(1)$ relaxation time for the natural additive and multiplicative reversibilizations of the systematic scan dynamics. Compared with our entropy result, the variance bound improves the dependence on the maximum interaction degree from exponential to quadratic and requires weaker assumptions on the distribution. As concrete applications of our results, we establish optimal $O(\log n)$ mixing of the systematic scan dynamics for bounded-degree antiferromagnetic two-spin systems in the tree-uniqueness region and for the ferromagnetic $q$-state Potts model on square boxes in $\mathbb{Z}^2$ throughout its subcritical regime.