Random cluster dynamics for the Ising model is rapidly mixing

Explorer Random Cluster Dynamics for the Ising Model is Rapidly Mixing

Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences

We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons [19]. For the ferromagnetic Ising model with inverse temperature β on a convergent sequence of graphs {Gn} with limit graphon W we show fast mixing of the Glauber dynamics if βλ1(W ) < 1 and slow (torpid) mixing if βλ1(W ) > 1 (where λ1(W ) is the largest eigenvalu...

Rapid mixing for the non-critical random-cluster model on the square lattice

We consider the random-cluster model with parameters p ∈ (0, 1) and q ∈ N on finite boxes in the two-dimensional square lattice for non-critical values of p, that is p 6= pc = √ q/(1 + √ q). We prove that the spectral gap of the continuous-time heat-bath dynamics for this model with free or wired boundary conditions is bounded from below by a constant. The proof uses the standard technique of b...

Quasi-polynomial Mixing of Critical 2d Random Cluster Models

We study the Glauber dynamics for the random cluster (FK) model on the torus (Z/nZ) with parameters (p, q), for q ∈ (1, 4] and p the critical point pc. The dynamics is believed to undergo a critical slowdown, with its continuous-time mixing time transitioning from O(logn) for p 6= pc to a power-law in n at p = pc. This was verified at p 6= pc by Blanca and Sinclair, whereas at the critical p = ...

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 a...