237 research outputs found

    Swendsen-Wang Algorithm on the Mean-Field Potts Model

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    We study the qq-state ferromagnetic Potts model on the nn-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=2q=2, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) Θ(1)\Theta(1) for β<βc\beta<\beta_c, (ii) Θ(n1/4)\Theta(n^{1/4}) for β=βc\beta=\beta_c, (iii) Θ(logn)\Theta(\log n) for β>βc\beta>\beta_c, where βc\beta_c is the critical temperature for the ordered/disordered phase transition. In contrast, for q3q\geq 3 there are two critical temperatures 0<βu<βrc0<\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 nn-vertex complete graph satisfies: (i) Θ(1)\Theta(1) for β<βu\beta<\beta_u, (ii) Θ(n1/3)\Theta(n^{1/3}) for β=βu\beta=\beta_u, (iii) exp(nΩ(1))\exp(n^{\Omega(1)}) for βu<β<βrc\beta_u<\beta<\beta_{rc}, and (iv) Θ(logn)\Theta(\log{n}) for ββrc\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

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    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

    Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics

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    We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability p(0,1)p \in (0,1) and a cluster weight q>0q > 0. We establish that for every q1q\ge 1, the random-cluster Glauber dynamics mixes in optimal Θ(nlogn)\Theta(n\log n) steps on nn-vertex random graphs having a prescribed degree sequence with bounded average branching γ\gamma throughout the full high-temperature uniqueness regime p<pu(q,γ)p<p_u(q,\gamma). The family of random graph models we consider include the Erd\H{o}s--R\'enyi random graph G(n,γ/n)G(n,\gamma/n), and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on the Erd\H{o}s--R\'enyi random graphs that works for all qq in the full uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the maximum degree) for the Potts Glauber dynamics, in the same settings where our Θ(nlogn)\Theta(n \log n) bounds for the random-cluster Glauber dynamics apply. This reveals a significant computational advantage of random-cluster based algorithms for sampling from the Potts Gibbs distribution at high temperatures in the presence of high-degree vertices.Comment: 45 pages, 3 figure

    Fast Algorithms at Low Temperatures via Markov Chains

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    For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok\u27s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces

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

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    We consider the ferromagnetic qq-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 β\beta. Our analysis concerns the low-temperature regime β\beta \to \infty, in which this multi-spin system has qq 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 qq-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as β\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=2q=2, our results characterize the tunneling behavior of the Ising model on grid graphs.Comment: 13 figure
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