Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

• Published in: Probability theory and related fields Vol. 152; no. 3-4; pp. 509 - 557
• Main Authors: Borgs, Christian, Chayes, Jennifer T., Tetali, Prasad
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2012; Springer Nature B.V
• Subjects: Algorithms; Boundary conditions; Computer science; Dynamic tests; Dynamics; Economics; Equilibrium; Finance; Heat; Insurance; Lattices; Lower bounds; Management; Markov analysis; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Phase transitions; Probability; Probability theory; Probability Theory and Stochastic Processes; Quantitative Finance; Statistical physics; Statistics for Business; Studies; Temperature; Theoretical; Topological manifolds; Transition points; Upper bounds; Primary 60J10; Partition width; Contour representation; Heat bath; Pirogov–Sinai theory; Secondary 60K35; 68Q87
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-010-0329-0

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice —heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d -1 . This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L /(log L ) 2 .