Showing papers on "Potts model published in 2012"

The Gravity Dual of the Ising Model

Abstract: We evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the AdS radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can – with certain assumptions – be computed and equals the vacuum character of a minimal model CFT. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton’s constant G and the AdS radius l the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known CFT. For example, the partition function of pure Einstein gravity with G = 3l equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E6, respectively.

Approximating the partition function of the ferromagnetic Potts model

Abstract: We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q-state Potts model when q > 2. Specifically, we show that the partition function is hard for the complexity class #RHPi under approximation-preserving reducibility. Thus, it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems, including that of determining the number of independent sets in a bipartite graph. Our proof exploits the first-order phase transition of the “random cluster” model, which is a probability distribution on graphs that is closely related to the q-state Potts model.

Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

Abstract: We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice \({\mathbb{Z}^{d}}\)—heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in Ld-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2.

Logarithmic observables in critical percolation

Abstract: Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q → 1 limit of the Q-state Potts model, and analyzing the underlying SQ symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q → 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d = 2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by carrying out extensive Monte Carlo simulations.

Glauber Dynamics for the mean-field Potts Model

Abstract: We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\beta \beta_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $\beta \uparrow \beta_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $\beta_s$. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.

Potts model based on a Markov process computation solves the community structure problem effectively.

Abstract: The Potts model is a powerful tool to uncover community structure in complex networks. Here, we propose a framework to reveal the optimal number of communities and stability of network structure by quantitatively analyzing the dynamics of the Potts model. Specifically we model the community structure detection Potts procedure by a Markov process, which has a clear mathematical explanation. Then we show that the local uniform behavior of spin values across multiple timescales in the representation of the Markov variables could naturally reveal the network's hierarchical community structure. In addition, critical topological information regarding multivariate spin configuration could also be inferred from the spectral signatures of the Markov process. Finally an algorithm is developed to determine fuzzy communities based on the optimal number of communities and the stability across multiple timescales. The effectiveness and efficiency of our algorithm are theoretically analyzed as well as experimentally validated.

Community structure detection based on Potts model and network's spectral characterization

Abstract: The Potts model was used to uncover community structure in complex networks. However, it could not reveal much important information such as the optimal number of communities and the overlapping nodes hidden in networks effectively. Differently from the previous studies, we established a new framework to study the dynamics of Potts model for community structure detection by using the Markov process, which has a clear mathematic explanation. Based on our framework, we showed that the local uniform behavior of spin values could naturally reveal the hierarchical community structure of a given network. Critical topological information regarding the optimal community structure could also be inferred from spectral signatures of the Markov process. A two-stage algorithm to detect community structure is developed. The effectiveness and efficiency of the algorithm has been theoretically analyzed as well as experimentally validated.

Replica inference approach to unsupervised multiscale image segmentation.

Abstract: We apply a replica-inference-based Potts model method to unsupervised image segmentation on multiple scales. This approach was inspired by the statistical mechanics problem of ``community detection'' and its phase diagram. Specifically, the problem is cast as identifying tightly bound clusters (``communities'' or ``solutes'') against a background or ``solvent.'' Within our multiresolution approach, we compute information-theory-based correlations among multiple solutions (``replicas'') of the same graph over a range of resolutions. Significant multiresolution structures are identified by replica correlations manifest by information theory overlaps. We further employ such information theory measures (such as normalized mutual information and variation of information), thermodynamic quantities such as the system entropy and energy, and dynamic measures monitoring the convergence time to viable solutions as metrics for transitions between various solvable and unsolvable phases. Within the solvable phase, transitions between contending solutions (such as those corresponding to segmentations on different scales) may also appear. With the aid of these correlations as well as thermodynamic measures, the phase diagram of the corresponding Potts model is analyzed at both zero and finite temperatures. Optimal parameters corresponding to a sensible unsupervised segmentations appear within the ``easy phase'' of the Potts model. Our algorithm is fast and shown to be at least as accurate as the best algorithms to date and to be especially suited to the detection of camouflaged images.

Estimating the granularity coefficient of a Potts-Markov random field within an MCMC algorithm

Abstract: This paper addresses the problem of estimating the Potts parameter B 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 B requires computing the intractable normalizing constant of the Potts model In the proposed MCMC method the estimation of B is conducted using a likelihood-free Metropolis-Hastings algorithm Experimental results obtained for synthetic data show that estimating B jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of B On the other hand, assuming that the value of B is known can degrade estimation performance significantly if this value is incorrect To illustrate the interest of this method, the proposed algorithm is successfully applied to real bidimensional SAR and tridimensional ultrasound images

Logarithmic observables in critical percolation

Abstract: Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q = 1 limit of the Q-state Potts model, and analyzing the underlying S_Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q = 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d=2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by extensive Monte-Carlo simulations.

Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model

Abstract: We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z2 domain symmetry breaking. Each loop receives a non-local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully packed, we analyze in detail the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q ≠ 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Cortical free-association dynamics: distinct phases of a latching network.

Abstract: A Potts associative memory network has been proposed as a simplified model of macroscopic cortical dynamics, in which each Potts unit stands for a patch of cortex, which can be activated in one of $S$ local attractor states. The internal neuronal dynamics of the patch is not described by the model, rather it is subsumed into an effective description in terms of graded Potts units, with adaptation effects both specific to each attractor state and generic to the patch. If each unit, or patch, receives effective (tensor) connections from $C$ other units, the network has been shown to be able to store a large number $p$ of global patterns, or network attractors, each with a fraction $a$ of the units active, where the critical load ${p}_{c}$ scales roughly like ${p}_{c}\ensuremath{\approx}C{S}^{2}/a\mathrm{ln}(1/a)$ (if the patterns are randomly correlated). Interestingly, after retrieving an externally cued attractor, the network can continue jumping, or latching, from attractor to attractor, driven by adaptation effects. The occurrence and duration of latching dynamics is found through simulations to depend critically on the strength of local attractor states, expressed in the Potts model by a parameter $w$. Here we describe with simulations and then analytically the boundaries between distinct phases of no latching, of transient and sustained latching, deriving a phase diagram in the plane $w\ensuremath{-}T$, where $T$ parametrizes thermal noise effects. Implications for real cortical dynamics are briefly reviewed in the conclusions.

Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model

Abstract: We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q eq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.

Phase separation and interface structure in two dimensions from field theory

Abstract: We study phase separation in two dimensions in the scaling limit below criticality. The general form of the magnetization profile as the volume goes to infinity is determined exactly within the field theoretical framework which explicitly takes into account the topological nature of the elementary excitations. The result known for the Ising model from its lattice solution is recovered as a particular case. In the asymptotic infrared limit the interface behaves as a simple curve characterized by a Gaussian passage probability density. The leading deviation, due to branching, from this behavior is also derived and its coefficient is determined for the Potts model. As a byproduct, for random percolation we obtain the asymptotic density profile of a spanning cluster conditioned to touch only the left half of the boundary.

Glauber Dynamics for the Mean-Field Potts Model

Abstract: We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β s (q) strictly lower than the critical β c (q) for uniqueness of the thermodynamic limit. The dynamical critical β s (q) is the spinodal point marking the onset of metastability. We prove that when β<β s (q) the mixing time is asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n 4/3. For β>β s (q) the mixing time is exponentially large in n. Furthermore, as β↑β s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n −2/3) around β s . These results form the first complete analysis of mixing around the critical dynamical temperature—including the critical power law—for a model with a first order phase transition.

Critical manifold of the kagome-lattice Potts model

Abstract: Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B?G; we call B a basis of G. We introduce a two-parameter graph polynomial PB(q, v) that depends on B and its embedding in G. The algebraic curve PB(q, v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = eK ? 1, defined on G. This curve predicts the phase diagram not only in the physical ferromagnetic regime (v > 0), but also in the antiferromagnetic (v 0 the accuracy of the predicted critical coupling vc is of the order 10?4 or 10?5 for the six-edge basis, and improves to 10?6 or 10?7 for the largest basis studied (with 36 edges).This article is part of ?Lattice models and integrability?, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Corner free energies and boundary effects for Ising, Potts and fully packed loop models on the square and triangular lattices

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 , where the coefficients αk and βk 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 → 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.

Growth laws and self-similar growth regimes of coarsening two-dimensional foams: transition from dry to wet limits.

Abstract: We study the topology and geometry of two-dimensional coarsening foam with an arbitrary liquid fraction. To interpolate between the dry limit described by von Neumann's law and the wet limit described by Marqusee's equation, the relevant bubble characteristics are the Plateau border radius and a new variable: the effective number of sides. We propose an equation for the individual bubble growth rate as the weighted sum of the growth through bubble-bubble interfaces and through bubble-Plateau border interfaces. The resulting prediction is successfully tested, without an adjustable parameter, using extensive bidimensional Potts model simulations. The simulations also show that a self-similar growth regime is observed at any liquid fraction, and they also determine how the average size growth exponent, side number distribution, and relative size distribution interpolate between the extreme limits. Applications include concentrated emulsions, grains in polycrystals, and other domains with coarsening that is driven by curvature.

Monte Carlo study of the triangular Blume-Capel model under bond randomness.

Abstract: The effects of bond randomness on the universality aspects of a two-dimensional (d = 2) Blume-Capel model embedded in the triangular lattice are discussed. The system is studied numerically in both its first- and second-order phase-transition regimes by a comprehensive finite-size scaling analysis for a particularly suitable value of the disorder strength. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the two-dimensional (2D) random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs again to the same universality class. Although the first finding reinforces the scenario of strong universality in the 2D Ising model with quenched disorder, the second is in difference from the critical behavior, emerging under randomness, in the cases of the ex-first-order transitions of the Potts model. Finally, our results verify previous renormalization-group calculations on the Blume-Capel model with disorder in the crystal-field coupling.

Loop Models, Random Matrices and Planar Algebras

Abstract: We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.

Critical manifold of the kagome-lattice Potts model

Abstract: Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B of G; we call B a basis of G. We introduce a two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = exp(K)-1, defined on G. This curve predicts the phase diagram both in the ferromagnetic (v>0) and antiferromagnetic (v<0) regions. For larger bases B the approximations become increasingly accurate, and we conjecture that P_B(q,v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P_B(q,v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker-Kadanoff phase. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wu's approach. The polynomial predictions are in excellent agreement with numerical computations. For v>0 the accuracy of the predicted critical coupling v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to 10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).

Statistical Mechanics on Isoradial Graphs

Abstract: Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.

Geometrical properties of the Potts model during the coarsening regime.

Abstract: We study the dynamic evolution of geometric structures in a polydegenerate system represented by a q-state Potts model with nonconserved order parameter that is quenched from its disordered into its ordered phase. The numerical results obtained with Monte Carlo simulations show a strong relation between the statistical properties of hull perimeters in the initial state and during coarsening: The statistics and morphology of the structures that are larger than the averaged ones are those of the initial state, while the ones of small structures are determined by the curvature-driven dynamic process. We link the hull properties to the ones of the areas they enclose. We analyze the linear von Neumann-Mullins law, both for individual domains and on the average, concluding that its validity, for the later case, is limited to domains with number of sides around 6, while presenting stronger violations in the former case.

Discontinuous phase transition in a dimer lattice gas.

Abstract: I study a dimer model on the square lattice with nearest neighbor exclusion as the only interaction. Detailed simulations using tomographic entropic sampling show that as the chemical potential is varied, there is a strongly discontinuous phase transition, at which the particle density jumps by about 18% of its maximum value, 1/4. The transition is accompanied by the onset of orientational order, to an arrangement corresponding to the {1/2, 0, 1/2} structure identified by Phares et al. [Physica B 409, 1096 (2011)] in a dimer model with finite repulsion at fixed density. Using finite-size scaling and Binder's cumulant, the expected scaling behavior at a discontinuous transition is verified in detail. The discontinuous transition can be understood qualitatively given that the model possesses eight equivalent maximum-density configurations, so that its coarse-grained description corresponds to that of the q = 8 Potts model.