Abstract
We study the Swendsen–Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen–Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen–Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from “unmixed” to “well mixed.” In particular, we show that at high enough temperatures the Swendsen–Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^{d}$ has cutoff at time $\frac{d}{2}(-\log (1-\gamma ))^{-1}\log n$, where $\gamma (\beta )$ is the spectral gap of the infinite-volume dynamics.
Citation
Danny Nam. Allan Sly. "Cutoff for the Swendsen–Wang dynamics on the lattice." Ann. Probab. 47 (6) 3705 - 3761, November 2019. https://doi.org/10.1214/19-AOP1344
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