Potts Model Research Articles

Finitary codings for gradient models and a new graphical representation for the six‐vertex model

AbstractIt is known that the Ising model on at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction to being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume‐order large deviation estimate for the energy. Results in the same spirit are shown for the Potts model, the so‐called beach model, and the six‐vertex model. We also introduce a coupling between the six‐vertex model with and a new Edwards–Sokal type graphical representation of it, which we believe is of independent interest.

Dynamic magnetic properties and multicritical phase diagram of the spin-1/2 Ashkin-Teller model under a time dependent external field

The spin-1/2 Ashkin-Teller model is studied via a Glauber type of stochastic process within the mean field theory. The thermal variation and the time evolution of the magnetizations under a time dependent external field are presented. The dynamic phase transitions caused by thermal variation and by the action of the time dependent external field are investigated, obtaining continuous phase transitions and first order phase transitions between metastable and stable states. In the phase diagram of the system, an arrow scheme is presented in order to illustrate the direction in which the dynamic phase transitions occur.

High pressure hydrogen and the Potts model on a triangular lattice

We present Monte Carlo studies and analysis of the frustrated antiferromagnetic Potts model of a triangular lattice. This Potts model shows a remarkably rich range of structures, and striking similarities to the high pressure phases of hydrogen which are typified by hexagonal close packed layered structures []. There are four known H2 molecular phases, all of which are isostructural to within the resolution of x-ray diffraction. Experimentally, the phase lines have been mapped by spectroscopy, which cannot reveal the structure. Study by density functional theory (DFT) has suggested a large number of candidate structures, based on the hexagonal-close packing of H2 molecules. The Potts model exhibits structures similar to DFT candidate hydrogen phases I, II and III: the range of different Potts model structures suggests that the hydrogen system in the ‘phase II’ region, may exhibit more than a single phase. It also suggests reorientational excitations which may be detectable in spectroscopy.

Phase diagrams and magnetization curves of the mixed Ashkin–Teller model including metastable and unstable states

Phase diagrams and magnetization curves of the mixed Ashkin–Teller model including metastable and unstable states

Multiple metastable states in an off-lattice Potts model

The interactions between a group of components are commonly studied in several areas of science (social science, biology, material science, complex dynamical systems, among others) using the methods of thermodynamics and statistical mechanics. In this work we study the properties of the recently proposed off-lattice, two-dimensional Potts model (Davis et al., 2014), originally motivated by the dynamics of agent opinions, and which is described by a Hamiltonian obtained by a maximum entropy inference procedure. We performed microcanonical and canonical Monte Carlo simulations of the first-order phase transition in the model, revealing a caloric curve with metastable regions. Furthermore, we report a “switching” behavior between multiple metastable states. We also note that the thermodynamics of the model has striking similarities with systems having long-range interactions, even though the interactions are short-ranged.

The Effect of Scoring Factor for Leiden Algorithm

Leiden algorithm is a widely utilized algorithm to cluster network graphs. It divides the specified network into smaller clusters. The clusters are relatively dense networks of vertices. In the process, the networks are divided based on quality factors. In this study, we compare the result of the Leiden algorithm with changing quality factors, namely Modularity and Constant Potts Model (CPM). For our analysis, we used 3×3 knight graph. Our investigation is completed for resolutions from 0.1 to 4.0 for Modularity and from 0.1 to 1.0 for CPM. The maximum quality scores are 0.9 and 0.59375 for Modularity and CPM respectively. The continuous decrease in the quality was recorded for both cases with respect to the increasing resolution. Both scoring factors are followed similar trends, but CPM has a relatively rapid division of the specified graph.

Modelling of Microstructure Evolution during Laser Processing of Intermetallic Containing Ni-Al Alloys

There is a growing interest in laser melting processes, e.g., for metal additive manufacturing. Modelling and numerical simulation can help to understand and control microstructure evolution in these processes. However, standard methods of microstructure simulation are generally not suited to model the kinetic effects associated with rapid solidification in laser processing, especially for material systems that contain intermetallic phases. In this paper, we present and employ a tailored phase-field model to demonstrate unique features of microstructure evolution in such systems. Initially, the problem of anomalous partitioning during rapid solidification of intermetallics is revisited using the tailored phase-field model, and the model predictions are assessed against the existing experimental data for the B2 phase in the Ni-Al binary system. The model is subsequently combined with a Potts model of grain growth to simulate laser processing of polycrystalline alloys containing intermetallic phases. Examples of simulations are presented for laser processing of a nickel-rich Ni-Al alloy, to demonstrate the application of the method in studying the effect of processing conditions on various microstructural features, such as distribution of intermetallic phases in the melt pool and the heat-affected zone. The computational framework used in this study is envisaged to provide additional insight into the evolution of microstructure in laser processing of industrially relevant materials, e.g., in laser welding or additive manufacturing of Ni-based superalloys.

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble hbox {CLE}_{kappa '} for kappa ' in (4, 8) that is drawn on an independent gamma -LQG surface for gamma ^2=16/kappa '. The results are similar in flavor to the ones from our companion paper dealing with hbox {CLE}_{kappa } for kappa in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the hbox {CLE}_{kappa '} in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a hbox {CLE}_{kappa '} independently into two colors with respective probabilities p and 1-p. This description was complete up to one missing parameter rho . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and kappa '. It shows in particular that hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} are related via a continuum analog of the Edwards-Sokal coupling between hbox {FK}_q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if q=4cos ^2(4pi / kappa '). This provides further evidence for the long-standing belief that hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} represent the scaling limits of hbox {FK}_q percolation and the q-Potts model when q and kappa ' are related in this way. Another consequence of the formula for rho (p,kappa ') is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

Gibbs measures of Potts model on Cayley trees: A survey and applications

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc.

Spatially clustered regression

Spatial regression and geographically weighted regression models have been widely adopted to capture the effects of auxiliary information on a response variable of interest over a region. In contrast, relationships between response and auxiliary variables are expected to exhibit complex spatial patterns in many applications. This paper proposes a new approach for spatial regression, called spatially clustered regression, to estimate possibly clustered spatial patterns of the relationships. We combine K-means-based clustering formulation and penalty function motivated from a spatial process known as Potts model for encouraging similar clustering in neighboring locations. We provide a simple iterative algorithm to fit the proposed method, scalable for large spatial datasets. Through simulation studies, the proposed method demonstrates its superior performance to existing methods even under the true structure does not admit spatial clustering. Finally, the proposed method is applied to crime event data in Tokyo and produces interpretable results for spatial patterns. The R code is available at https://github.com/sshonosuke/SCR.

Monte Carlo examination of first-order phase transitions in a system with many independent order parameters: Three-dimensional Ashkin-Teller model.

A Monte Carlo (MC) computer experiment for the analysis of first-order temperature-driven phase transitions in a system with one or many independently behaving order parameters is presented using the example of the three-dimensional (3D) Ashkin-Teller model, one of the important reference systems in statistical physics showing a rich and complex phase diagram. The properties of a number of quantities, such as magnetization, three types of cumulants, the internal energy, and its histogram, are exploited. The Lee and Kosterlitz concept proposed for strong first-order phase transitions in systems with one independent order parameter is significantly expanded to obtain results with comparable error bars in reasonable computation times at an arbitrary amount of latent heat. The proposed computer MC experiment uses parallel processing and both the Metropolis and recently formulated cluster algorithms. Arbitrarily weak to strong first-order phase transitions in the phase diagram region with ferromagnetic interactions are investigated and the latent heat associated with individual degrees of freedom is carefully computed. In the discussion of results, the behavior of our 3D system between that of the mean-field and that of the 2D one is bracketed and the role of the Potts point is clarified.

Computer simulation of microstructure development in powder-bed additive manufacturing with crystallographic texture

Parts produced via laser powder-bed fusion additive manufacturing exhibit complex microstructures that depend on processing variables and often vary widely in crystallographic texture and grain morphology. The need to understand, predict, and control these microstructural variations motivates the development of modeling tools capable of accurately predicting LPBF microstructures. Monte Carlo (MC) Potts models have been employed to successfully model the formation of grain structures in additively manufactured parts but have lacked the ability to simulate crystallographic texture. We present an extension of the MC Potts model that assigns an orientation to each grain and penalizes growth of solid into the fusion zone based on proximity of the nearest 〈100〉 crystal direction to the local temperature gradient direction. This allows for crystallographically selective growth to drive texture formation during the development of the solidification microstructure in each melt track. LPBF builds of alloy 718 with a unidirectional scan pattern provided microstructures with substantial variations in grain size, grain morphology, and texture. These distinctive albeit atypical microstructures were used to validate the simulation method, i.e. good agreement was obtained between the simulated and experimental grain shapes and textures.

Chaotic renormalization flow in the Potts model induced by long-range competition.

A proper description of spin glass remains a hard subject in theoretical physics and is considered to be closely related to the emergence of chaos in the renormalization group (RG) flow. Previous efforts concentrate on models with either complicated or nonrealistic interactions in order to achieve this chaotic behavior. Here we find that the commonly used Potts model with long-range interaction could do the job nicely in a large parameter regime as long as the competition between the ferromagnetic and antiferromagnetic interaction is maintained. With this simplicity, the appearance of chaos is observed to sensitively depend on the detailed network structure: the parity of bond number in a branch of the basic RG substituting unit; chaos only emerges for even numbers of bonds. These surprising and universal findings may shed light on the study of spin glass.

PPalign: optimal alignment of Potts models representing proteins with direct coupling information

BackgroundTo assign structural and functional annotations to the ever increasing amount of sequenced proteins, the main approach relies on sequence-based homology search methods, e.g. BLAST or the current state-of-the-art methods based on profile Hidden Markov Models, which rely on significant alignments of query sequences to annotated proteins or protein families. While powerful, these approaches do not take coevolution between residues into account. Taking advantage of recent advances in the field of contact prediction, we propose here to represent proteins by Potts models, which model direct couplings between positions in addition to positional composition, and to compare proteins by aligning these models. Due to non-local dependencies, the problem of aligning Potts models is hard and remains the main computational bottleneck for their use.MethodsWe introduce here an Integer Linear Programming formulation of the problem and PPalign, a program based on this formulation, to compute the optimal pairwise alignment of Potts models representing proteins in tractable time. The approach is assessed with respect to a non-redundant set of reference pairwise sequence alignments from SISYPHUS benchmark which have lowest sequence identity (between 3% and 20%) and enable to build reliable Potts models for each sequence to be aligned. This experimentation confirms that Potts models can be aligned in reasonable time (1'37'' in average on these alignments). The contribution of couplings is evaluated in comparison with HHalign and independent-site PPalign. Although Potts models were not fully optimized for alignment purposes and simple gap scores were used, PPalign yields a better mean F_1 score and finds significantly better alignments than HHalign and PPalign without couplings in some cases.ConclusionsThese results show that pairwise couplings from protein Potts models can be used to improve the alignment of remotely related protein sequences in tractable time. Our experimentation suggests yet that new research on the inference of Potts models is now needed to make them more comparable and suitable for homology search. We think that PPalign’s guaranteed optimality will be a powerful asset to perform unbiased investigations in this direction.

Theory and simulation for equilibrium glassy dynamics in cellular Potts model of confluent biological tissue

Glassy dynamics in a confluent monolayer is indispensable in morphogenesis, wound healing, bronchial asthma, and many others; a detailed theoretical framework for such a system is, therefore, important. Vertex-model (VM) simulations have provided crucial insights into the dynamics of such systems, but their nonequilibrium nature makes theoretical development difficult. The cellular Potts model (CPM) of confluent monolayers provides an alternative model for such systems with a well-defined equilibrium limit. We combine numerical simulations of the CPM and an analytical study based on one of the most successful theories of equilibrium glass, the random first-order transition theory, and develop a comprehensive theoretical framework for a confluent glassy system. We find that the glassy dynamics within the CPM is qualitatively similar to that in the VM. Our study elucidates the crucial role of geometric constraints in bringing about two distinct regimes in the dynamics, as the target perimeter P_{0} is varied. The unusual sub-Arrhenius relaxation results from the distinctive interaction potential arising from the perimeter constraint in such systems. The fragility of the system decreases with increasing P_{0} in the low-P_{0} regime, whereas the dynamics is independent of P_{0} in the other regime. The rigidity transition, found in the VM, is absent within the CPM; this difference seems to come from the nonequilibrium nature of the former. We show that the CPM captures the basic phenomenology of glassy dynamics in a confluent biological system via comparison of our numerical results with existing experiments on different systems.

Analyzing Recrystallization Behavior of Heterogeneous Structures Single-Phase Al Alloys

In this research, mixed structures are created so that they have regions where dislocations accumulate and a region that is, to some extent, free from dislocations. These heterogeneous structures that bring together both soft and hard domains in a single sample are presumed to produce a fine recrystallized grain and recovered large grain structures if they are subsequently heated, leading to limited recrystallization. In the present work, the 3D Potts model has been modified to tailor and model complex problems such as heterogeneous structure recrystallization. This is done by introducing a limited fraction of nucleating grains to the structure where the highest geometrically necessary dislocation density distributions develop in the rolled layers at early simulation time as well as by controlling structure evolution in the undeformed structure layer using grain boundary energy criteria. Additionally, the designed microstructures were investigated using electron backscatter diffraction measurements and an in-situ heating stage performed on material with various degrees of deformation.

Periodic Gibbs measures for the antiferromagnetic Potts model on a Cayley tree of order k

Periodic Gibbs measures for the antiferromagnetic Potts model on a Cayley tree of order k

Effect of Quenched Nonmagnetic Impurities on the Phase Transitions in a Three-Dimensional Potts Model

Effect of Quenched Nonmagnetic Impurities on the Phase Transitions in a Three-Dimensional Potts Model

Quantum criticality in many-body parafermion chains

We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of Z_3Z3 parafermion or u(1)_6u(1)6 conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of kk-state clock type models.

Numerical evidence of superuniversality of the two-dimensional and three-dimensional random quantum Potts models

The random q-state quantum Potts model is studied on hypercubic lattices in dimensions 2 and 3 using the numerical implementation of the Strong Disorder Renormalization Group introduced by Kovacs and Igl{\'o}i [Phys. Rev. B 82, 054437 (2010)]. Critical exponents $\nu$, d f and $\psi$ at the Infinite Disorder Fixed Point are estimated by Finite-Size Scaling for several numbers of states q between 2 and 50. When scaling corrections are not taken into account, the estimates of both d f and $\psi$ systematically increase with q. It is shown however that q-dependent scaling corrections are present and that the exponents are compatible within error bars, or close to each other, when these corrections are taking into account. This provides evidence of the existence of a super-universality of all 2D and 3D random Potts models.