Paper Info
Title
Dynamics for the Mean-field Random-cluster Model.
Abstract
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 $(\lambda_s,\lambda_S)$.
Year
DOI
Venue
2015
10.4230/LIPIcs.APPROX-RANDOM.2015.528
APPROX-RANDOM
Field
DocType
Volume
Spin-½,Combinatorics,Random graph,Exponential function,Markov chain,Slowdown,Mean field theory,Spanning tree,Mathematics,Lambda
Conference
40
Citations 
PageRank 
References 
0
0.34
4
Authors
2
Name
Order
Citations
PageRank
Antonio Blanca1149.74
Alistair Sinclair21506308.40