Showing papers on "Potts model published in 2013"

Gibbs Measures on Cayley Trees

Abstract: Properties of a Group Representation of the Cayley Tree Ising Model on Cayley Tree Ising Type Models with Competing Interactions Information Flow on Trees The Potts Model The Solid-on-Solid Model Models with Hard Constraints Potts Model with Countable Set of Spin Values Models with Uncountable Set of Spin Values Contour Arguments on Cayley Trees Other Models.

Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications

Abstract: We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as the limit Q->1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c->0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O(-2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.

Generic, hierarchical framework for massively parallel Wang-Landau sampling.

Abstract: We introduce a parallel Wang-Landau method based on the replica-exchange framework for Monte Carlo simulations. To demonstrate its advantages and general applicability for simulations of complex systems, we apply it to different spin models including spin glasses, the Ising model, and the Potts model, lattice protein adsorption, and the self-assembly process in amphiphilic solutions. Without loss of accuracy, the method gives significant speed-up and potentially scales up to petaflop machines.

Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications

Abstract: We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n → 0 limit of the O(n) model, and percolation as the limit Q → 1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c → 0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O( − 2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.

Estimating the Granularity Coefficient of a Potts-Markov Random Field Within a Markov Chain Monte Carlo Algorithm

Abstract: This paper addresses the problem of estimating the Potts parameter β jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on β requires computing the intractable normalizing constant of the Potts model. In the proposed MCMC method, the estimation of β is conducted using a likelihood-free Metropolis-Hastings algorithm. Experimental results obtained for synthetic data show that estimating β jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of β. On the other hand, choosing an incorrect value of β can degrade estimation performance significantly. To illustrate the interest of this method, the proposed algorithm is successfully applied to real bidimensional SAR and tridimensional ultrasound images.

A Survey and Comparison of Discrete and Continuous Multi-label Optimization Approaches for the Potts Model

Abstract: We present a survey and a comparison of a variety of algorithms that have been proposed over the years to minimize multi-label optimization problems based on the Potts model. Discrete approaches based on Markov Random Fields as well as continuous optimization approaches based on partial differential equations can be applied to the task. In contrast to the case of binary labeling, the multi-label problem is known to be NP hard and thus one can only expect near-optimal solutions. In this paper, we carry out a theoretical comparison and an experimental analysis of existing approaches with respect to accuracy, optimality and runtime, aimed at bringing out the advantages and short-comings of the respective algorithms. Systematic quantitative comparison is done on the Graz interactive image segmentation benchmark. This paper thereby generalizes a previous experimental comparison (Klodt et al. 2008) from the binary to the multi-label case.

Factor models on locally tree-like graphs

Abstract: We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T, and study the existence of the free energy density ϕ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute ϕ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T. In the special case that T has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.

Description of the translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree

Abstract: For the $q$-state Potts model on a Cayley tree of order $k\geq 2$ it is well-known that at sufficiently low temperatures there are at least $q+1$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is $2^{q}-1$. We prove that there are $[q/2]$ (where $[a]$ is the integer part of $a$) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

Abstract: In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

Transfer matrix computation of critical polynomials for two-dimensional Potts models

Abstract: In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK − 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction–deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible. We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction–deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.

Phase transitions of the q-state Potts model on multiply-laced Sierpinski gaskets

Abstract: We present an exact solution of the q-state Potts model on a class of generalized Sierpinski fractal lattices. The model is shown to possess an ordered phase at low temperatures and a continuous transition to the high temperature disordered phase at any q ≥ 1. Multicriticality is observed in the presence of a symmetry-breaking field. Exact renormalization group analysis yields the phase diagram of the model and a complete set of critical exponents at various transitions.

Comparison of Swendsen‐Wang and heat‐bath dynamics

Abstract: We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the two-dimensional Potts models at all non-critical temperatures. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 520–535, 2013

Analysis of stability of community structure across multiple hierarchical levels

Abstract: The analysis of stability of community structure is an important problem for scientists from many fields. Here, we propose a new framework to reveal hidden properties of community structures by quantitatively analyzing the dynamics of the Potts model. Specifically we model the Potts procedure of community structure detection by a Markov process, which has a clear mathematical explanation. Critical topological information regarding multivariate spin configuration could also be inferred from the spectral significance of the Markov process. We test our framework on some example networks and find it does not have resolution limitation problems at all. Results have shown the model we proposed is able to uncover the hierarchical structure in different scales effectively and efficiently.

A Bayesian non-parametric Potts model with application to pre-surgical FMRI data

Abstract: The Potts model has enjoyed much success as a prior model for image segmentation. Given the individual classes in the model, the data are typically modeled as Gaussian random variates or as random variates from some other parametric distribution. In this manuscript we present a non-parametric Potts model and apply it to an FMRI study for the pre-surgical assessment of peritumoral brain activation. In our model we assume that the Z-score image from a patient can be segmented into activated, deactivated and null classes, or states. Conditional on the class, or state, the Z-scores are assumed to come from some generic distribution which we model non-parametrically using a mixture of Dirichlet process priors within the Bayesian framework. The posterior distribution of the model parameters is estimated with a Markov chain Monte Carlo algorithm and Bayesian decision theory is used to make the final classifications. Our Potts prior model includes two parameters, the standard spatial regularization parameter and a parameter that can be interpreted as the a priori probability that each voxel belong to the null, or background state, conditional on the lack of spatial regularization. We assume that both of these parameters are unknown, and jointly estimate them along with other model parameters. We show through simulation studies that our model performs on par, in terms of posterior expected loss, with parametric Potts models when the parametric model is correctly specified, and outperforms parametric models when the parametric model in misspecified.

Periodic Gibbs measures for the Potts model on the Cayley tree

Abstract: We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures.

Phase transitions for p-adic Potts model on the Cayley tree of order three

Abstract: In the present paper, we study a phase transition problem for the q-state p-adic Potts model over the Cayley tree of order three. We consider a more general notion of p-adic Gibbs measure which depends on parameter ρ∈Qp. Such a measure is called generalized p-adic quasi Gibbs measure. When ρ equals the p-adic exponent, then it coincides with the p-adic Gibbs measure. When ρ = p, then it coincides with the p-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of |ρ|p. Namely, in the first regime, one takes ρ = expp(J) for some J∈Qp, in the second one |ρ|p < 1. In each regime, we first find conditions for the existence of generalized p-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when |ρ|p,|q|p ≤ p−2 we prove the existence of a quasi phase transition. It turns out that if $\vert \rho \vert _{p}\lt \vert q-1\vert _{p}^{2}\lt 1$ and $\sqrt{-3}\in {\mathbb{Q}}_{p}$, then one finds the existence of the strong phase transition.

Potts model, parametric maxflow and k-submodular functions

Abstract: The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19,20]. It identifies a part of an optimal solution by running $k$ maxflow computations, where $k$ is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to $O(\log k)$ maxflow computations (or one {\em parametric maxflow} computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for {\em Tree Metrics}. We also show a connection to {\em $k$-submodular functions} from combinatorial optimization, and discuss {\em $k$-submodular relaxations} for general energy functions.

Potts Model, Parametric Maxflow and K-Submodular Functions

Abstract: The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19, 20]. It identifies a part of an optimal solution by running k maxflow computations, where k is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to O(log k) maxflow computations (or one parametric maxflow computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for Tree Metrics. We also show a connection to k-sub modular functions from combinatorial optimization, and discuss k-sub modular relaxations for general energy functions.

Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree

Abstract: For the Potts model with competing interactions, we describe the set of weakly periodic ground states corresponding to index-two normal divisors of the Cayley tree group representation. We also study some weakly periodic Gibbs measures.

Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model

Abstract: We analyze the generalized mean-field q-state Potts model which is obtained by replacing the usual quadratic interaction function in the meanfield Hamiltonian by a higher power z. We first prove a generalization of the known limit result for the empirical magnetization vector of Ellis and Wang [9] which shows that in the right parameter regime, the first-order phase-transition persists. Next we turn to the corresponding generalized fuzzy Potts model which is obtained by decomposing the set of the q possible spin-values into 1 < s < q classes and identifying the spins within these classes. In extension of earlier work [21] which treats the quadratic model we prove the following: The fuzzy Potts model with interaction exponent bigger than four (respectively bigger than two and smaller or equal four) is non-Gibbs if and only

On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models

Abstract: We consider general cyclic representations of the 6-vertex Yang-Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov-Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin's quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit.

Partial Optimality via Iterative Pruning for the Potts Model

Abstract: We propose a novel method to obtain a part of an optimal non-relaxed integral solution for energy minimization problems with Potts interactions, known also as the minimal partition problem. The method empirically outperforms previous approaches likeMQPBO and Kovtun’s method in most of our test instances and especially in hard ones. As a starting point our approach uses the solution of a commonly accepted convex relaxation of the problem. This solution is then iteratively pruned until our criterion for partial optimality is satisfied. Due to its generality our method can employ any solver for the considered relaxed problem.

Zero-temperature coarsening in the 2d Potts model

Abstract: We study the fate of the 2d kineticq-state Potts model after a sudden quench to zero temperature. Both ground states and complicated static states are reached with non-zero probabilities. These outcomes resemble those found in the quench of the 2d Ising model; however, the variety of static states in the q-state Potts model (with q 3) is much richer than in the Ising model, where static states are either ground or stripe states. Another possibility is that the Potts system gets trapped on a set of equal-energy blinker states, where a subset of spins can ip ad innitum ; these states are similar to those found in the quench of the 3d Ising model. The evolution towards the nal energy is also unusual|at long times, sudden and massive energy drops may occur that are accompanied by macroscopic reordering of the domain structure. This indeterminacy in the zero-temperature quench of the kinetic Potts model is at odds with basic predictions from the theory of phase-ordering kinetics. We also propose a continuum description of coarsening with more than two equivalent ground states. The resulting time- dependent Ginzburg{Landau equations reproduce the complex cluster patterns that arise in the quench of the kinetic Potts model.