Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

Abstract

The random-cluster model with parameters (p, q) is a random graph model that generalizes bond percolation (q = 1) and the Ising and Potts models (q ≥ 2). We study its Glauber dynamics on n × n boxes Λ n of the integer lattice graph Z 2 , where the model exhibits a sharp phase transition at p = p c (q). Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of Λ n , known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when q > 1 and p = p c (q), the Glauber dynamics on Λ n mixes in optimal O(n 2 log n) time. In this paper, we prove that this mixing time is polynomial in n for every boundary condition that is realizable as a configuration on Z 2 \ Λ n . We then use this to prove near-optimalÕ(n 2 ) mixing time for "typical" boundary conditions. As a complementary result, we construct classes of non-realizable (non-planar) boundary conditions inducing slow (stretched-exponential) mixing at p ≪ p c (q).We say e is a cut-edge in (Λ n , S t ) if the number of connected components in S t ∪ {e} and S t \ {e} differ. Under a boundary condition ξ, the property of e being a cut-edge is defined with respect to the augmented graph (Λ n , S ξ t ), where S ξ t adds external wirings between all pairs of vertices in the same element of ξ. This Markov chain converges to π Λn,p,q by construction, and we study its speed of convergence.A standard measure for quantifying the speed of convergence of a Markov chain is the mixing time, which is defined as the time until the dynamics is close (in total variation distance) to its stationary distribution, starting from a worst-case initial state. We say the dynamics is rapidly mixing if the mixing time is polynomial in |V |, and torpidly mixing when the mixing time is exponential in |V | ε for some ε > 0.The corresponding Glauber dynamics for the Ising/Potts model (which updates spins one at a time according to the spins of their neighbors), is by now quite well understood on finite regions of Z 2 . In the high-temperature region β < β c (corresponding to p < p c ) the Glauber dynamics has optimal mixing time Θ(n 2 log n) on boxes Λ n [28, 7, 2, 1]; moreover, this same asymptotic bound of Θ(n 2 log n) holds for every fixed boundary condition. These bounds follow as a consequence of the exponential decay of correlations of the model in the high-temperature regime, which holds even near the boundary for arbitrary Ising/Potts boundary conditions; this property is known as strong spatial mixing. In the low-temperature region β > β c the mixing time is exponential in n for free and periodic (toroidal) boundaries [35,6,17]. The more general problem of understanding the mixing time of the Glauber dynamics for other boundary conditions at low temperatures is a long-standing open problem, e.g., see [25,29].The FK-dynamics is quite powerful since the self-...

Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

“…The order in which the uniform variables are revealed is irrelevant and can be adaptive; this will allow us to reveal the boundary components. (For more details on the process of revealing random-cluster components under the monotone coupling, see below, as well as e.g., [6,8]. )…”

“…The following formalizes the notion that sparse boundary conditions are "close to free", and allows us to compare the induced mixing time on balls with sparse boundary to those with free boundary. Definition 15 (Definition 2.1 of [6]). For two boundary conditions (partitions) φ ≤ φ , define D(φ, φ ) := c(φ) − c(φ ) where c(φ) is the number of components in φ.…”

“…This is due, in part, to the fact that the random-cluster model presents highly non-local interactions: an update on an edge e t depends on the entire configuration ω t (E \{e t }). Indeed, the only other setting where the speed of convergence of FK-dynamics is well-understood via direct analysis is in square subsets of Z 2 [6,8,[26][27][28][29]. Other bounds to date have been obtained either indirectly, via comparison with global Markov chains using the results of [52,53] (and as a result, these bounds are off by polynomial factors), or by taking either p very small (e.g., under a Dobrushin-type condition) or very large, or q large.…”

Sampling from Potts on random graphs of unbounded degree via random-cluster dynamics

“…We compile here a number of definitions and results that formalize this idea. Definition 6.3 (Definition 2.1 from [6]). For two boundary conditions (partitions) φ ď φ 1 , define Dpφ, φ 1 q :" cpφq ´cpφ 1 q where cpφq is the number of components in φ.…”

“…As such, any sampling algorithm for the random-cluster model yields one for the ferromagnetic Potts model with essentially no computational overhead. This fact has led to significantly improved sampling algorithms for the Potts model in various low-temperature settings [2,3,7,8,12,13,20,34,39,41] and more generally, to a broad interest in dynamics for the random-cluster model [6,9,[21][22][23][24]26,28].…”