Showing papers on "Potts model published in 2011"
TL;DR: A smoothing method based on the log-sum exponential function is developed and indicates that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy.
Abstract: This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding duality-based approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.
182 citations
TL;DR: In this article, the enumeration of properly q-colored planar maps was studied and the associated generating function is algebraic when q 0,4 is of the form 2+2cos(j@p/m), for integers j and m, including the two integer values q=2 and q=3.
63 citations
TL;DR: In this paper, the Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree was shown to be tight for q ≥ 5.
Abstract: The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a nonbinary model establishing the Kesten–Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten–Stigum bound is not tight for the q-state Potts model when q ≥ 5. Moreover, we determine asymptotics for these reconstruction thresholds.
61 citations
TL;DR: This work provides a new interpolation scheme and uses 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, and shows that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T.
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 $\phi$, 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 $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ 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 $\phi$. 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.
59 citations
TL;DR: In this article, an approach modifying the standard Monte Carlo Potts model algorithm is presented, where modifications of the mobility of grain boundaries and their associated boundary junctions allow the simulation of the size effect observed in nanocrystalline grain growth.
58 citations
TL;DR: In this paper, structural properties of the q-color Potts field theory were studied for real values of q, which describes the scaling limit of the random cluster model, and it was shown that the number of independent n -point Potts spin correlators coincides with the independent n-point cluster connectivities and is given by generalized Bell numbers.
47 citations
TL;DR: In this paper, a family of phase transitions connecting phases of differing nontrivial topological order is described by explicitly constructing Hamiltonians of the Levin-Wen type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase.
Abstract: We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2 $+$ 1 dimensions in terms of doubled Chern-Simons theories.
44 citations
TL;DR: In this paper, size effects observed in nanocrystalline grain growth are modelled by attributing a specific energy and finite mobility to each structural feature of a polyhedral grain by considering grain growth as a dissipative process that is driven by the reduction in the Gibbs free interface, edge and vertex energy.
43 citations
TL;DR: In this article, the authors give efficient quantum algorithms to estimate the partition function of (i) the six-vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on quasi-2D square lattices and (iv) the lattice gauge theory on a 3D cubic lattice.
Abstract: We give efficient quantum algorithms to estimate the partition function of (i) the six-vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi-2D square lattice and (iv) the lattice gauge theory on a 3D square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced by Van den Nest et al (2009 Phys. Rev. A 80 052334) and extended here.
40 citations
TL;DR: An iteration-free weighted histogram method in terms of intensive variables that directly determines the inverse statistical temperature, β(S) = ∂S/∂E, with S the microcanonical entropy that provides insights into the nature of the underlying phase transitions via signatures in β( S) characteristic of finite size systems.
Abstract: We present an iteration-free weighted histogram method in terms of intensive variables that directly determines the inverse statistical temperature, βS = ∂S/∂E, with S the microcanonical entropy. The method eliminates iterative evaluations of the partition functions intrinsic to the conventional approach and leads to a dramatic acceleration of the posterior analysis of combining statistically independent simulations with no loss in accuracy. The synergistic combination of the method with generalized ensemble weights provides insights into the nature of the underlying phase transitions via signatures in βS characteristic of finite size systems. The versatility and accuracy of the method is illustrated for the Ising and Potts models.
39 citations
TL;DR: Strong evidence is presented for a previously unknown, "entropy-driven," finite-temperature phase transition to a partially ordered state in the Potts model on the Union-Jack lattice.
Abstract: We evaluate the thermodynamic properties of the 4-state antiferromagnetic Potts model on the Union-Jack lattice using tensor-based numerical methods. We present strong evidence for a previously unknown, "entropy-driven," finite-temperature phase transition to a partially ordered state. From the thermodynamics of Potts models on the diced and centered diced lattices, we propose that finite-temperature transitions and partially ordered states are ubiquitous on irregular lattices.
TL;DR: In this paper, it was shown 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 gaps of any single-spin 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 of the Swendsen-Wang process 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.
TL;DR: In this paper, the authors give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model and (iv) the Z_2 lattice gauge theory on a three-dimensional square topology.
Abstract: We give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a three-dimensional square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced in [Van den Nest et al., Phys. Rev. A 80, 052334 (2009)] and extended here.
TL;DR: In this article, a sharp threshold theorem is proved for box-crossing probabilities on the square lattice in the random-cluster model near the self-dual point psd(q) = √q∕(1+√q), the Ising model with external field, and the colored random clustering model.
Abstract: A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point psd(q)=√q∕(1+√q), the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.
TL;DR: It is proved the existence of a phase transition at a positive critical temperature by identifying a suitable interpolation structure and an extended variational principle together with a positive temperature second-moment analysis.
Abstract: We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.
Posted Content•
TL;DR: For the q-state ferromagnetic Potts model on trees with n vertices, the mixing time is Theta(1) in high temperatures, Theta (log n) in low temperatures and Theta n −1/4 at criticality as mentioned in this paper.
Abstract: The Swendsen-Wang dynamics is a Markov chain widely used by physicists to sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer, Frieze and Rue proved that on the complete graph K_n the mixing time of the chain is at most O(n^{1/2}) for all non-critical temperatures. In this paper we show that the mixing time is Theta(1) in high temperatures, Theta(log n) in low temperatures and Theta(n^{1/4}) at criticality. We also provide an upper bound of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on any tree with n vertices.
TL;DR: In this paper, the authors studied the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for non-integer q ≥ 1.
Abstract: We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes–Machta dynamics for the Fortuin–Kasteleyn random-cluster model, which generalizes the Swendsen–Wang dynamics for the q-state Potts ferromagnet to non-integer q≥1. We consider spatial dimension d=2 and 1.25≤q≤4 in steps of 0.25, on lattices up to 10242, and obtain estimates for the dynamic critical exponent z CM. We present evidence that when 1≤q≲1.95 the Ossola–Sokal conjecture z CM≥β/ν is violated, though we also present plausible fits compatible with this conjecture. We show that the Li–Sokal bound z CM≥α/ν is close to being sharp over the entire range 1≤q≤4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.
TL;DR: In this paper, the authors investigate the crossover from the pure short-range interacting model to the long-range interaction model and derive a formula for the critical temperature as a function of the strength of the longrange interaction.
Abstract: Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.
Journal Article•
TL;DR: This article proposed a conditional composite likelihood based on conditional prob- abilities of parts of the data given the rest for spatial lattice processes, in particular for Potts models, which generalizes the pseudo-likelihood of Besag.
Abstract: We propose a conditional composite likelihood based on conditional prob- abilities of parts of the data given the rest for spatial lattice processes, in particular for Potts models, which generalizes the pseudo-likelihood of Besag. Instead of using conditional probabilities of single pixels given the rest (like Besag), we use condi- tional probabilities of multiple pixels given the rest. We find that our maximum composite likelihood estimates (MCLE) are more efficient than maximum pseudo- likelihood estimates (MPLE) when the true parameter value of the Potts model is the phase transition parameter value. Our MCLE are not as efficient as maxi- mum likelihood estimates (MLE), but MCLE and MPLE can be calculated exactly, whereas MLE cannot, only approximated by Markov chain Monte Carlo.
TL;DR: In this paper, a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two is studied.
Abstract: In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12, No. 2, 141‐156] on phase transition for the Ising model to the Potts model setting.
TL;DR: In this article, the authors obtained long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations.
Abstract: We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q-state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model (Q=2) at temperature T, as well as a fully-packed O(n) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q, n or 1/T.
By using a carefully chosen expansion parameter, q << 1, all expansions turn out to be of the form \prod_{k=1}^\infty (1-q^k)^{\alpha_k + k \beta_k}, where the coefficients \alpha_k and \beta_k are periodic functions of k. Thanks to this periodicity property we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 <= q < 1.
We analyse in detail the q \to 1^- limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases.
Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen-Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r. These results are in turn compared with CFT predictions.
29 May 2011
TL;DR: This work shows that an accelerated first-order algorithm of Nesterov is well suited for the multi-label Potts model, due to its low memory requirements and its potential for massive parallelism.
Abstract: The Potts model is a well established approach to solve different multi-label problems. The classical Potts prior penalizes the total interface length to obtain regular boundaries. Although the Potts prior works well for many problems, it does not preserve fine details of the boundaries. In recent years, non-local regularizers have been proposed to improve different variational models. The basic idea is to consider pixel interactions within a larger neighborhood. This can for example be used to incorporate low-level segmentation into the regularizer which leads to improved boundaries. In this work we study such an extension for the multi-label Potts model. Due to the increased model complexity, the main challenge is the development of an efficient minimization algorithm. We show that an accelerated first-order algorithm of Nesterov is well suited for this problem, due to its low memory requirements and its potential for massive parallelism. Our algorithm allows us to minimize the non-local Potts model with several hundred labels within a few minutes. This makes the non-local Potts model applicable for computer vision problems with many labels, such as multi-label image segmentation and stereo.
TL;DR: In this article, the authors investigate the phase diagram of a three-component system of particles on a one-dimensional filled lattice, or equivalently of a one dimensional three-state Potts model, with reflection asymmetric mean-field interactions.
Abstract: We investigate the phase diagram of a three-component system of particles on a one-dimensional filled lattice, or equivalently of a one-dimensional three-state Potts model, with reflection asymmetric mean-field interactions. The three types of particles are designated as A, B and C. The system is described by a grand canonical ensemble with temperature T and chemical potentials TλA, TλB and TλC. We find that for λA = λB = λC the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature . For other values of the chemical potentials the system has a unique equilibrium state. As is the case for the canonical ensemble for this ABC model, the grand canonical ensemble is the stationary measure satisfying detailed balance for a natural dynamics. We note that , where Tc is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities rA = rB = rC = 1/3.
TL;DR: The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method and an intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble.
Abstract: The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.
TL;DR: In this article, the censoring inequality for Glauber dynamics on monotone spins systems such as the Ising model was shown to fail for proper colorings of a graph, and for lazy transpositions on permutations.
Abstract: Peres and Winkler proved a ‘censoring' inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known whether the censoring property holds in other natural settings such as the Potts model.
TL;DR: In this paper, the partition function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature were studied and shown to become dense everywhere in the complex q-plane outside the limiting curve.
Abstract: We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m×n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve \(\mathcal{B}_{\infty}(\mathrm{sq})\) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.
02 Jan 2011
TL;DR: In this article, the authors consider the problem of finding the probability that a path along the oriented bonds passes sequentially fixed sites i and j, and show that the probability in the predecessors problem for two sites separated by large distance r decreases as p ( r ) ∼ r −3/4.
Abstract: We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is to find the probability that a path along the oriented bonds passes sequentially fixed sites i and j . The conformal field theory for the Potts model predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in the predecessors problem for two sites separated by large distance r decreases as P ( r ) ∼ r −3/4 . If sites i and j are nearest neighbors on the square lattice, the probability P (1) = 5 / 16 can be found from the analytical theory developed for the sandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on the spanning tree states that P (1) is the probability for the LERW started at i to reach the neighboring site j . By analogy with the self-avoiding walk, P (1) can be called the return probability. Extensive Monte-Carlo simulations confirm the theoretical predictions.
TL;DR: A growth equation for individual grains is suggested by considering the interactions of the nearest-neighbor grains, which indicates that the average grain growth rate of a given topological class depends on the difference between the number of faces.
Abstract: A growth equation for individual grains is suggested by considering the interactions of the nearest-neighbor grains, which indicates that the average grain growth rate of a given topological class depends on the difference between the number of faces and the average number of faces of the nearest neighbors. For the convenience of its application, a practical equation is also proposed. They have been verified by the data of β-titanium grains, dry 3D foams, Monte Carlo Potts model simulation, vertex model simulation, and surface evolver simulation.
TL;DR: The V-polynomial as mentioned in this paper generalizes Noble and Welsh's W-Polynomial by incorporating vertex weights and adapting contraction to accommodate them, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields.