Swendsen-Wang Algorithm on the Mean-Field Potts Model

We study the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case $q=2$, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) $\Theta(1)$ for $\beta<\beta_c$, (ii) $\Theta(n^{1/4})$ for $\beta=\beta_c$, (iii) $\Theta(\log n)$ for $\beta>\beta_c$, where $\beta_c$ is the critical temperature for the ordered/disordered phase transition. In contrast, for $q\geq 3$ there are two critical temperatures $0<\beta_u<\beta_{rc}$ that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts model on the $n$-vertex complete graph satisfies: (i) $\Theta(1)$ for $\beta<\beta_u$, (ii) $\Theta(n^{1/3})$ for $\beta=\beta_u$, (iii) $\exp(n^{\Omega(1)})$ for $\beta_u<\beta<\beta_{rc}$, and (iv) $\Theta(\log{n})$ for $\beta\geq\beta_{rc}$. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.Comment: To appear in Random Structures & Algorithm

Swendsen-Wang Algorithm on the Mean-Field Potts Model

We study the q-state ferromagnetic Potts model on the n-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0=beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over

Random-Cluster Dynamics in Z2\mathbb{Z}^2

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a $O(n^2\log n)$ upper bound for the mixing time at all values of the model parameter $p$ except the critical point $p=p_c(q)$, and for all values of the second model parameter $q\ge 1$. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in $\mathbb{Z}^2$. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense

Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $\beta \to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.Comment: 13 figure

Low-temperature Ising dynamics with random initializations

It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be polynomially fast. However, such phenomena have largely evaded rigorous analysis, as most tools in the analysis of Markov chain mixing times are tailored to worst-case initializations. In this paper, we establish this conjectured behavior for the classical case of the Ising model on an $N$-vertex torus in $\mathbb Z^d$. Namely, we show that the mixing time for the Glauber dynamics, initialized in a $\frac 12$-$\frac 12$ mixture of the all-plus and all-minus configurations, is $O(N\log N)$ in all dimensions, at all temperatures below the critical one. We also give a matching lower bound, showing that this is optimal. A key innovation in our analysis is the introduction of the notion of "weak spatial mixing within a phase", an adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to imply rapid mixing of the Glauber dynamics from the above random initialization, and that it holds for the Ising model at all low temperatures and in all dimensions. As byproducts of our analysis, we also prove rapid mixing of the Glauber dynamics restricted to the plus phase, and the Glauber dynamics on a box in $\mathbb Z^d$ with plus boundary conditions, when initialized from the all-plus configuration.Comment: 32 pages, 3 figure