237 research outputs found
Swendsen-Wang Algorithm on the Mean-Field Potts Model
We study the -state ferromagnetic Potts model on the -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 , the
Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and
showed that the mixing time satisfies: (i) for ,
(ii) for , (iii) for
, where is the critical temperature for the
ordered/disordered phase transition. In contrast, for there are two
critical temperatures that are relevant. We prove that
the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts
model on the -vertex complete graph satisfies: (i) for
, (ii) for , (iii)
for , and (iv)
for . 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
Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics
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 and a cluster weight . We establish
that for every , the random-cluster Glauber dynamics mixes in optimal
steps on -vertex random graphs having a prescribed degree
sequence with bounded average branching throughout the full
high-temperature uniqueness regime .
The family of random graph models we consider include the Erd\H{o}s--R\'enyi
random graph , 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 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
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
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
We consider the ferromagnetic -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
. Our analysis concerns the low-temperature regime ,
in which this multi-spin system has 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 -state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case , our results characterize the
tunneling behavior of the Ising model on grid graphs.Comment: 13 figure
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