Dynamics for the Mean-field Random-cluster Model

The random-cluster model is a unifying framework for studying random graphs, spin systems in physics and random spanning trees. The model is closely related to, though much more general than the classical Ising and Potts models, but its dynamics are much less well understood. In this paper we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime $(\lambda_s,\lambda_S)$ of the model parameter $\lambda$ in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is $\Theta(\log n)$ if $\lambda \not\in [\lambda_s,\lambda_S]$, and $\exp(\Omega(\sqrt{n}))$ when $\lambda \in (\lambda_s,\lambda_S)$. These results hold for all values of the second model parameter $q > 1$. Thus, we obtain the first analysis of a dynamics for the random-cluster model for values of $q$ other than the already well understood special case $q=2$ (which corresponds to the Ising model) over almost the full range of values of $\lambda$. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in $(\lambda_s,\lambda_S)$.

[1]  Christian Borgs,et al.  Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point , 2010, ArXiv.

[2]  Béla Bollobás,et al.  Random Graphs , 1985 .

[3]  V. Beara.  The self-dual point of the two-dimensional random-cluster model is critical for q > 1 , 2011 .

[4]  Mario Ullrich,et al.  Rapid mixing of Swendsen-Wang and single-bond dynamics in two dimensions , 2012, ArXiv.

[5]  G. Grimmett,et al.  The random-cluster model on the complete graph , 1996 .

[6]  Jonathan Machta,et al.  GRAPHICAL REPRESENTATIONS AND CLUSTER ALGORITHMS II , 1998 .

[7]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[8]  Mario Ullrich,et al.  Swendsen-Wang Is Faster than Single-Bond Dynamics , 2012, SIAM J. Discret. Math..

[9]  Paul Cuff,et al.  Glauber Dynamics for the Mean-Field Potts Model , 2012, 1204.4503.

[10]  Svante Janson,et al.  Susceptibility in subcritical random graphs , 2008, 0806.0252.

[11]  Eric Vigoda,et al.  Swendsen‐Wang algorithm on the mean‐field Potts model , 2015, APPROX-RANDOM.

[12]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[13]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[14]  Tomasz Luczak,et al.  The phase transition in the cluster‐scaled model of a random graph , 2006, Random Struct. Algorithms.

[15]  Remco van der Hofstad,et al.  Random Graphs and Complex Networks , 2016, Cambridge Series in Statistical and Probabilistic Mathematics.

[16]  Asaf Nachmias,et al.  A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics , 2011 .

[17]  Alistair Sinclair,et al.  Dynamics for the Mean-field Random-cluster Model , 2014, APPROX-RANDOM.

[18]  Geoffrey Grimmett.  The Random-Cluster Model , 2002, math/0205237.

[19]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[20]  Qi Ge,et al.  A Graph Polynomial for Independent Sets of Bipartite Graphs , 2009, Combinatorics, Probability and Computing.

[21]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[22]  Eric Vigoda,et al.  Swendsen-Wang Algorithm on the Mean-Field , 2015 .

[23]  R. Durrett.  Probability: Theory and Examples , 1993 .

[24]  Martin Dyer,et al.  Mixing properties of the Swendsen–Wang process on the complete graph and narrow grids , 2000 .

[25]  F. Martinelli.  Rapid mixing of Swendsen-Wang dynamics in two dimensions , 2012 .

[26]  Mark Huber,et al.  A bounding chain for Swendsen‐Wang , 2003, Random Struct. Algorithms.

[27]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[28]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[29]  Mark Jerrum,et al.  The Swendsen-Wang process does not always mix rapidly , 1997, STOC '97.

[30]  Alan M. Frieze,et al.  Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[31]  G. Grimmett,et al.  Probability and random processes , 2002 .

[32]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.