Showing papers on "Potts model published in 2007"

Limited resolution in complex network community detection with Potts model approach

Abstract: According to Fortunato and Barthelemy, modularity-based community detection algorithms have a resolution threshold such that small communities in a large network are invisible. Here we generalize their work and show that the q-state Potts community detection method introduced by Reichardt and Bornholdt also has a resolution threshold. The model contains a parameter by which this threshold can be tuned, but no a priori principle is known to select the proper value. Single global optimization criteria do not seem capable for detecting all communities if their size distribution is broad.

Fast mixing for independent sets, colorings, and other models on trees

Abstract: We study the mixing time of the Glauber dynamics for general spin systems on the regular tree, including the Ising model, the hard-core model (independent sets), and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper (Martinelli, Sinclair, and Weitz, Tech. Report UCB//CSD-03-1256, Dept. of EECS, UC Berkeley, July 2003) in the context of the Ising model, for establishing mixing time O(nlog n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition. We also discuss applications of our framework to reconstruction problems on trees. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 A preliminary version of this paper appeared in Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, January 2004. This work was done while the author was visiting the Departments of EECS and Statistics, University of California, Berkeley, supported in part by a Miller Visiting Professorship.

The Complexity of Ferromagnetic Ising with Local Fields

Abstract: We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomized approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterize the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logically defined subclass of nP previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the $q$-state Potts model with local external magnetic fields and $q>2$ is complete for all of nP with respect to approximation-preserving reductions.

On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree

Abstract: In the present paper we consider countable state $p$-adic Potts model on the Cayley tree. A construction of $p$-adic Gibbs measures which depends on weights $\l$ is given, and an investigation of such measures is reduced to examination of an infinite-dimensional recursion equation. Studying of the derived equation under some condition on weights, we prove absence of the phase transition. Note that the condition does not depend on values of the prime $p$, and an analogues fact is not true when the number of spins is finite. For homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one $p$-adic Gibbs measure $\m_\l$. The boundedness of the measure is also established. Moreover, continuous dependence the measure $\m_\l$ on $\l$ is proved. At the end we formulate one limit theorem for $\m_\l$.

On p-adic Gibbs measures of the countable state Potts model on the Cayley tree

Abstract: In this paper we consider the countable state p-adic Potts model on the Cayley tree. A construction of p-adic Gibbs measures which depends on weights λ is given, and an investigation of such measures is reduced to the examination of an infinite-dimensional recursion equation. By studying the derived equation under some condition concerning weights, we prove the absence of a phase transition. Note that the condition does not depend on values of the prime p, and the analogous fact is not true when the number of spins is finite. For the homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one p-adic Gibbs measure μλ. The boundedness of the measure is also established. Moreover, the continuous dependence of the measure μλ on λ is proved. At the end we formulate a one limit theorem for μλ.

SLE in the three-state Potts model - a numerical study

Abstract: The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with kappa=3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with kappa=10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramm's formula. The results are in support of our hypothesis.

Microcanonical approach to the simulation of first-order phase transitions.

Abstract: A simple microcanonical strategy for the simulation of first-order phase transitions is proposed. At variance with flat-histogram methods, there is no iterative parameters optimization nor long waits for tunneling between the ordered and the disordered phases. We test the method in the standard benchmark: the Q-states Potts model (Q=10 in two dimensions and Q=4 in D=3). We develop a cluster algorithm for this model, obtaining accurate results for systems with more than 10(6) spins.

Potts Glass on Random Graphs

Abstract: We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular we argue that in the colorable region the one-step replica symmetry breaking solution is stable towards more steps of replica symmetry breaking.

Schramm–Loewner evolution in the three-state Potts model—a numerical study

Abstract: The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is Schramm–Loewner evolution (SLE) with κ = 3. We hypothesize that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with κ = 10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramm's formula. The results are in support of our hypothesis.

Liquid crystals in two dimensions: first-order phase transitions and nonuniversal critical behavior.

Abstract: Liquid crystals in two dimensions undergo a first-order isotropic-to-quasi-nematic transition, provided the particle interactions are sufficiently "sharp and narrow." This implies phase coexistence between isotropic and quasi-nematic domains, separated by interfaces. The corresponding line tension is determined and shown to be very small, giving rise to strong interface fluctuations. When the interactions are no longer "sharp and narrow," the transition becomes continuous, with nonuniversal critical behavior obeying hyperscaling and approximately resembling the two-dimensional Potts model.

Numerical tests of local scale invariance in ageing q-state Potts models

Abstract: Much effort has been spent over the last years to achieve a coherent theoretical description of ageing as a non-linear dynamics process. Long supposed to be a consequence of the slow dynamics of glassy systems only, ageing phenomena could also be identified in the phase-ordering kinetics of simple ferromagnets. As a phenomenological approach Henkel et al. developed a group of local scale transformations under which two-time autocorrelation and response functions should transform covariantly. This work is to extend previous numerical tests of the predicted scaling functions for the Ising model by Monte Carlo simulations of two-dimensional q-state Potts models with q=3 and 8, which, in equilibrium, undergo temperature-driven phase transitions of second and first order, respectively.

Dynamic critical behavior of the worm algorithm for the Ising model.

Abstract: We study the dynamic critical behavior of the worm algorithm for the two- and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than the Swendsen-Wang algorithm for simulating the two-point function of the three-dimensional Ising model.

Ferromagnetic phase transition for the spanning-forest model (q-->0 limit of the Potts model) in three or more dimensions.

Abstract: We present Monte Carlo simulations of the spanning-forest model (q-->0 limit of the ferromagnetic Potts model) in spatial dimensions d=3, 4, 5. We show that, in contrast to the two-dimensional case, the model has a ferromagnetic second-order phase transition at a finite positive value w(c). We present numerical estimates of w(c) and of the thermal and magnetic critical exponents. We conjecture that the upper critical dimension is 6.

On the Ising model for the simple cubic lattice

Abstract: The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for ...

Increasing entanglement through engineered disorder in the random Ising chain

Abstract: The ground-state entanglement entropy between block of sites in the random Ising chain is studied by means of the Von Neumann entropy. We show that in presence of strong correlations between the disordered couplings and local magnetic fields the entanglement increases and becomes larger than in the ordered case. The different behavior with respect to the uncorrelated disordered model is due to the drastic change of the ground state properties. The same result holds also for the random three-state quantum Potts model.

Critical Behavior of the Chayes–Machta–Swendsen–Wang Dynamics

Abstract: We study 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 model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z >or= alpha/nu is close to but probably not sharp in d = 2 and is far from sharp in d = 3, for all q. The conjecture z >or= beta/nu is false (for some values of q) in both d = 2 and d = 3.

Fusion operators in the generalized τ(2)-model and root-of-unity symmetry of the XXZ spin chain of higher spin*

Abstract: We construct the fusion operators in the generalized τ(2)-model using the fused L-operators, and verify the fusion relations with the truncation identity. The algebraic Bethe-ansatz discussion is conducted on two special classes of τ(2) which include the superintegrable chiral Potts model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and demonstrate that the sl2-loop-algebra symmetry exists for the root-of-unity XXZ spin chain with a higher spin, where the evaluation parameters for the symmetry algebra are identified by the explicit Fabricius–McCoy current for the Bethe states. Parallels are also drawn to the comparison with the superintegrable chiral Potts model.

Potts model with competing interactions on the Cayley tree: The contour method

Abstract: We consider the Potts model with three spin values and with competing interactions of radius r = 2 on the Cayley tree of order k = 2. We completely describe the ground states of this model and use the contour method on the tree to prove that this model has three Gibbs measures at sufficiently low temperatures.

Competitive nucleation and the Ostwald rule in a generalized Potts model with multiple metastable phases

Abstract: We introduce a simple nearest-neighbor spin model with multiple metastable phases, the number and decay pathways of which are explicitly controlled by the parameters of the system. With this model, we can construct, for example, a system which evolves through an arbitrarily long succession of metastable phases. We also construct systems in which different phases may nucleate competitively from a single initial phase. For such a system, we present a general method to extract from numerical simulations the individual nucleation rates of the nucleating phases. The results show that the Ostwald rule, which predicts which phase will nucleate, must be modified probabilistically when the new phases are almost equally stable. Finally, we show that the nucleation rate of a phase depends, among other things, on the number of other phases accessible from it.

First-Order Phase Transition in Potts Models with Finite-Range Interactions

Abstract: We consider the Q-state Potts model on Z d , Q≥ 3, d≥ 2, with Kac ferromagnetic interactions and scaling parameter γ. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for γ small enough there is a value of the temperature at which coexist Q+1 Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for d = 2, Q = 3, in contrast with the case of nearest-neighbor interactions for which available results indicate a second order phase transition. Putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction.

Long-term ordering kinetics of the two-dimensional q-state Potts model.

Abstract: We studied the nonequilibrium dynamics of the q-state Potts model in the square lattice, after a quench to subcritical temperatures. By means of a continuous time Monte Carlo algorithm (nonconserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For q>4 we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system gets stuck in highly symmetric nonequilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in highly disordered metastable states with finite lifetime, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency, are analyzed in detail.

Potts Model with two Competing Binary Interactions

Abstract: The Potts model on a Cayley tree in the presence of competing two binary interactions and magnetic field is considered. We exactly solve a problem of phase transitions for the model,namely we calculate critical surface such that there is a phase transition above it,and a single Gibbs state found elsewhere.

Using the P n Potts model with learning methods to segment live cell images

Abstract: We present a segmentation method for live cell images, using graph cuts and learning methods. The images used here are particularly challenging because of the shared grey-level distributions of cells and background, which only differ by their textures, and the local imprecision around cell borders. We use the Pn Potts model recently presented by Kohli et al. [9]: functions on higher-order cliques of pixels are included into the traditional Potts model, allowing us to account for local texture features, and to find the optimal solution efficiently. We use learning methods to define the potential functions used in the Pn Potts model. We present the model and the learning methods we used, and compare our segmentation results with similar work in cytometry. While our method performs similarly, it requires little manual tuning and thus is straightforward to adapt to other images.

Positive association in the fractional fuzzy Potts model

Abstract: A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph G obtained in two steps: first a subgraph of G is chosen according to a random cluster measure Φ p,q , and then a spin (±1) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever q ≥ 1, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of Haggstrom [Ann. Appl. Probab. 9 (1999) 1149-1159] and Haggstrom and Schramm [Stochastic Process. Appl. 96 (2001) 213-242].

Corner transfer matrices in statistical mechanics

Abstract: Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensional models. They can be a very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model.

First-order transition features of the triangular Ising model with nearest- and next-nearest-neighbor antiferromagnetic interactions

Abstract: We implement a new and accurate numerical entropic scheme to investigate the first-order transition features of the triangular Ising model with nearest-neighbor ( J nn ) and next-nearest-neighbor ( J nnn ) antiferromagnetic interactions in ratio R = J nn / J nnn = 1 . Important aspects of the existing theories of first-order transitions are briefly reviewed, tested on this model, and compared with previous work on the Potts model. Using lattices with linear sizes L = 30 , 40 , … , 100 , 120 , 140 , 160 , 200 , 240 , 360 and 480 we estimate the thermal characteristics of the present weak first-order transition. Our results improve the original estimates of Rastelli et al. and verify all the generally accepted predictions of the finite-size scaling theory of first-order transitions, including transition point shifts, thermal, and magnetic anomalies. However, two of our findings are not compatible with current phenomenological expectations. The behavior of transition points, derived from the number-of-phases parameter, is not in accordance with the theoretically conjectured exponentially small shift behavior and the well-known double Gaussian approximation does not correctly describe higher correction terms of the energy cumulants. It is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections.