## The Annals of Probability

### Cutoff for the Swendsen–Wang dynamics on the lattice

#### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 3705-3761.

Dates
Revised: January 2019
First available in Project Euclid: 2 December 2019

https://projecteuclid.org/euclid.aop/1575277340

Digital Object Identifier
doi:10.1214/19-AOP1344

Mathematical Reviews number (MathSciNet)
MR4038041

#### Citation

Nam, Danny; Sly, Allan. Cutoff for the Swendsen–Wang dynamics on the lattice. Ann. Probab. 47 (2019), no. 6, 3705--3761. doi:10.1214/19-AOP1344. https://projecteuclid.org/euclid.aop/1575277340

#### References

• [1] Blanca, A., Caputo, P., Sinclair, A. and Vigoda, E. (2018). Spatial mixing and non-local Markov chains. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms 1965–1980. SIAM, Philadelphia, PA.
• [2] Borgs, C., Chayes, J. T., Frieze, A., Kim, J. H., Tetali, P., Vigoda, E. and Vu, V. H. (1999). Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. In 40th Annual Symposium on Foundations of Computer Science (New York, 1999) 218–229. IEEE Computer Soc., Los Alamitos, CA.
• [3] Borgs, C., Chayes, J. T. and Tetali, P. (2012). Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point. Probab. Theory Related Fields 152 509–557.
• [4] Bubley, R. and Dyer, M. (1997). Path coupling: A technique for proving rapid mixing in Markov chains. In Proc. 38th Annual Sympos. on Foundations of Computer Science 223–231.
• [5] Cooper, C., Dyer, M. E., Frieze, A. M. and Rue, R. (2000). Mixing properties of the Swendsen–Wang process on the complete graph and narrow grids. J. Math. Phys. 41 1499–1527.
• [6] Cooper, C. and Frieze, A. M. (1999). Mixing properties of the Swendsen–Wang process on classes of graphs. Random Structures Algorithms 15 242–261.
• [7] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
• [8] Du, J., Zheng, B. and Wang, J. S. (2006). Dynamic critical exponents for Swendsen–Wang and Wolff algorithms obtained by a nonequilibrium relaxation method. J. Stat. Mech. 2006. P05004.
• [9] Duminil-Copin, H., Gagnebin, M., Harel, M., Manolescu, I. and Manolescu, I. (2016). Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$. Available at arXiv:1611.09877.
• [10] Duminil-Copin, H., Sidoravicius, V. and Tassion, V. (2017). Continuity of the phase transition for planar random-cluster and Potts models with $1\leq q\leq 4$. Comm. Math. Phys. 349 47–107.
• [11] Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 2009–2012.
• [12] Fortuin, C. M. (1971). On the random-cluster model. Doctoral thesis.
• [13] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
• [14] Galanis, A., Štefankovič, D. and Vigoda, E. (2019). Swendsen–Wang algorithm on the mean-field Potts model. Random Structures Algorithms 54 82–147.
• [15] Gheissari, R. and Lubetzky, E. (2018). Mixing times of critical two-dimensional Potts models. Comm. Pure Appl. Math. 71 994–1046.
• [16] Gheissari, R. and Lubetzky, E. (2018). The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions. Electron. J. Probab. 23 Paper No. 57, 30.
• [17] Gheissari, R., Lubetzky, E. and Peres, Y. (2018). Exponentially slow mixing in the mean-field Swendsen–Wang dynamics. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms 1981–1988. SIAM, Philadelphia, PA.
• [18] Gore, V. K. and Jerrum, M. R. (1999). The Swendsen–Wang process does not always mix rapidly. J. Stat. Phys. 97 67–86.
• [19] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
• [20] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
• [21] Guo, H. and Jerrum, M. (2018). Random cluster dynamics for the Ising model is rapidly mixing. Ann. Appl. Probab. 28 1292–1313.
• [22] Levin, D. A., Luczak, M. J. and Peres, Y. (2010). Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability. Probab. Theory Related Fields 146 223–265.
• [23] Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times. 2nd ed. Amer. Math. Soc., Providence, RI.
• [24] Long, Y., Nachmias, A. and Peres, Y. (2007). Mixing time power laws at criticality. In Proc. 48th Annual IEEE Sympos. on Foundations of Computer Science 205–214.
• [25] Lubetzky, E. and Sly, A. (2013). Cutoff for the Ising model on the lattice. Invent. Math. 191 719–755.
• [26] Lubetzky, E. and Sly, A. (2014). Cutoff for general spin systems with arbitrary boundary conditions. Comm. Pure Appl. Math. 67 982–1027.
• [27] Lubetzky, E. and Sly, A. (2015). An exposition to information percolation for the Ising model. Ann. Fac. Sci. Toulouse Math. (6) 24 745–761.
• [28] Lubetzky, E. and Sly, A. (2016). Information percolation and cutoff for the stochastic Ising model. J. Amer. Math. Soc. 29 729–774.
• [29] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 93–191. Springer, Berlin.
• [30] Miller, J. and Peres, Y. (2012). Uniformity of the uncovered set of random walk and cutoff for lamplighter chains. Ann. Probab. 40 535–577.
• [31] Ossola, G. and Sokal, A. D. (2004). Dynamic critical behavior of the Swendsen–Wang algorithm for the three-dimensional Ising model. Nuclear Phys. B 691 259–291.
• [32] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
• [33] Swendsen, R. H. and Wang, J. S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58 86–88.
• [34] Ullrich, M. (2013). Comparison of Swendsen–Wang and heat-bath dynamics. Random Structures Algorithms 42 520–535.
• [35] Ullrich, M. (2014). Rapid mixing of Swendsen–Wang dynamics in two dimensions. Dissertationes Math. 502 64.