Markov chain Monte Carlo algorithm for the mean field random-cluster model

The random-cluster model has emerged as a unifying framework for studying random graphs, physical spin systems, and electrical networks [6]. Given a base graph G = (V,E), the random-cluster model consists of a probability distribution π over subsets of E, where p ∈ [0, 1] and q > 0 are real parameters determining the bond and cluster weights respectively. For q ≥ 1, there is a value of p = pc at which the random-cluster model undergoes a phase transition. The heat-bath dynamics is a local Markov chain used to sample from π. For G = Kn, the mean field case, we show that the heat-bath dynamics mixes in O(n log n) steps when q > 1 and p < min{pc, 2 n}. Our result provides the first upper bounds for the mixing time for non-integer values of q, and it is tight up to a logarithmic factor. For some specific cases where an upper bound was previously known, we improve such bounds by at least a O(n) factor. Our result also provides the first polynomial bound for the mixing time of the Swendsen-Wang dynamics when q ≥ 3 and 1/3 n < p < 2 n . 1 Problem and Motivation The random-cluster model was created by Fortuin and Kasteleyn in the late 1960’s as a unifying framework for studying random graphs, physical spin systems, and electrical networks [6]. Let G = (V,E) be a graph, p ∈ [0, 1], and q > 0. The configurations of the random-cluster model are all subgraphs H=(V,E0) of G, which we identify with their edge set E0 ⊆ E. The probability of a configuration E0 is: π(E0 ⊆ E) = p|E0|(1− p)|E|−|E0|qc(E0) ZRC where c(E0) denotes the number of connected components of H = (V,E0), and ZRC is a normalizing constant called the partition function. The special case when q = 1 has been widely studied in the framework of random graphs [1]. In particular when G is the complete graph with n vertices (denoted Kn), the randomcluster model reduces to the standard ErdősRényi G(n, p) model where a configuration is obtained by adding each edge with probability p independently. When q < 1, configurations with fewer connected components (“clusters”) are favored whereas the opposite happens when q > 1. The random-cluster model is a generalization of the Ising (q = 2) and Potts models (q ≥ 3 with q ∈ Z), two fundamental models for ferromagnetism in statistical physics [11, 20]. Much of the physical theory of the Ising and Potts models can be understood in the context of the randomcluster model. In the late 1980’s, interest in the random-cluster model resurged [5, 17], and it became a central tool for modeling and understanding the ferromagnet and its phase transition [8]. In addition, because the random-cluster model is a model for random graphs which takes into account connectivity properties, it is also useful in the study of electrical and other networks [8]. 1.1 The heat-bath dynamics The heat-bath dynamics is a local Markov chain over the space of random-cluster configurations Ω = {E0 : E0 ⊆ E}. Given a configuration E0 ⊆ E, the heat-bath dynamics performs the following random step: 1. Choose an edge e ∈ E uniformly at random. 2. With probability pe the new configuration is E0 ∪ {e}, where pe is the conditional probability of E0 ∪ {e} determined by the state of the other edges: pe = π(E0 ∪ {e}) π(E0 ∪ {e}) + π(E0 \ {e}) . Otherwise, the new configuration is E0\{e} :