Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.     

On Large Deviation Properties of Erdös–Rényi Random Graphs

We show that large deviation properties of Erdös-Rényi random graphs can be derived from the free energy of the q-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to ln q is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the q 1 limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.     

Translation invariant Gibbs states in theq-state Potts model

We describe the set of all translation invariant Gibbs states in theq-state Potts model for the case ofq large enough and the other parameters to be arbitrary.     

Residual entropy of the one-dimensional Ising antiferromagnet with general spin

We have calculated the ground-state degeneracy and the concominant residual entropy of the one-dimensional Ising antiferromagnet in a critical magnetic field. We demonstrate that the results obtained can be related to the ground-state properties of the q-state Potts antiferromagnet in the corresponding external field.     

Approaches of the percolation theory and the free energy of dislocation clusters

The feasibility of describing ordered dislocation networks in crystals in terms of approaches inherent in the percolation theory is investigated theoretically. The free energy of dislocation networks forming clusters is calculated by the methods of the percolation theory using the Ising and Potts models and the Onsager solution.     

Agreement percolation and phase coexistence in some Gibbs systems

We extend some relations between percolation and the dependence of Gibbs states on boundary conditions known for Ising ferromagnets to other systems and investigate their general validity: percolation is defined in terms of the agreement of a configuration with one of the ground states of the system. This extension is studied via examples and counterexamples, including the antiferromagnetic Ising and hard-core models on bipartite lattices, Potts models, and many-layered Ising and continuum Widom-Rowlinson models. In particular our results on the hard square lattice model make rigorous observations made by Hu and Mak on the basis of computer simulations. Moreover, we observe that the (naturally defined) clusters of the Widom-Rowlinson model play (for the WR model itself) the same role that the clusters of the Fortuin-Kasteleyn measure play for the ferromagnetic Potts models. The phase transition and percolation in this system can be mapped into the corresponding liquid-vapor transition of a one-component fluid.     

Cyclic period-3 window in antiferromagnetic potts and Ising models on recursive lattices

The magnetic properties of the antiferromagnetic Potts model with two-site interaction and the antiferromagnetic Ising model with three-site interaction on recursive lattices have been studied. A cyclic period-3 window has been revealed by the recurrence relation method in the antiferromagnetic Q-state Potts model on the Bethe lattice (at Q < 2) and in the antiferromagnetic Ising model with three-site interaction on the Husimi cactus. The Lyapunov exponents have been calculated, modulated phases and a chaotic regime in the cyclic period-3 window have been found for one-dimensional rational mappings determined the properties of these systems.     

Potts model on complex networks

We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of p = 1, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.Received: 3 December 2003, Published online: 17 February 2004PACS: 05.10.-a Computational methods in statistical physics and nonlinear dynamics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 87.18.Sn Neural networks     

The semi-infinite Potts model: A new low temperature phase

We consider the semi-infiniteq-state Potts model in the many component limitq. Both mean field theory and the Migdal-Kadanoff renormalization group scheme are used to obtain an approximate surface free energy. Both methods predict a new low temperature phase where the bulk is ordered while the free surface is disordered.     

Phase Transition for the Ising Model on the Critical Lorentzian Triangulation

The ferromagnetic Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of the Peierls contour method. The critical temperature is shown to be constant a.s.     

Fractal geometry of critical Potts clusters

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. A new method for thermalization of spin systems is presented. The method allows a speed up of an order of magnetization for large lattices. We also show snapshots of the Potts clusters for different values of q, which clearly illustrate the fact that the clusters become more compact as q increases, and that this affects the fractal dimensions in a monotonic way. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponents.Received: 2 June 2003, Published online: 9 September 2003PACS: 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) - 05.45.Df Fractals - 75.10.-b General theory and models of magnetic ordering - 75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.)     

The potts model on bethe lattices

We analyze the properties of theq-state ferromagnetic Potts model for realq. The nature of the phase transition at the critical point is first-order forq2, and second-order forq=2. The random-bond percolation limitq1, and its second-order-like transition, are not related to the previous behaviour since they arise from non-stable phases of the system. It is suggested that this property characterizes the model on high-dimensional lattices, too.Supported by MPI and CNR     

Dynamics of the infinite-ranged Potts model

We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.     

Thermodynamic limit of theq-state Potts-Hopfield model with infinitely many patterns

We prove the almost sure convergence of the free energy and of the overlap order parameters in aq-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the systemN like (/lnq) lnN. We study the limiting behavior of the extremal states of the model that are the measures induced on the Gibbs measures by the overlap parameters.     

Exact results on the random Potts model

We present a number of exact results on the random-bond,q-state Potts model. The quenched model on any finite planar graph or lattice is shown to obey a duality relation for general type of bond-randomness. In the annealed case, the solution of the model reduces to that of the regular (nonrandom) Potts model on the corresponding lattice. Explicit knowledge of the critical parameters of theq-state Potts model in two dimensions allows us to evaluate exactly the phase diagram of the annealed model on the square, triangular and honeycomb lattices. We discuss the behavior near the (random) critical point and comment on the relationship between the quenched and annealed systems. The exact phase diagram of the annealed system is obtained for the bond-diluted model and the spin-glass model with and without dilutions.Work supported in part by NSF grant No. DMR-78-18808     

Renormalization group approach to the surface and defect critical behavior in the potts model

A simple real-space renormalization group method with two-terminal clusters is used to treat the critical behavior of Potts ferromagnet with free surface and defect plane on the same footing both for square and cubic lattices. For a square lattice, quite different critical behaviors are found for the cases of line defect and free surface. Whenq is larger than three, like the case ofa line type defect in ‘diamond’ hierarchical lattice, the order parameter on a defect line increases discontinuously at the bulk critical point if the defect interaction is sufficiently strong. This behavior, on the contrary, does not occur on the surface of a semi-infinite plane. For a cubic lattice, the phase diagram and renormalization group flow properties are obtained explicitly for bothq=1 (bond percolation) andq=2 (Ising model). In both cases, our calculations whow that the critical behavior on the surface of a semi-infinite system belongs to a different universality class from the critical behavior on the defect plane of a bulk system.     

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

We consider the scaling limit of the two-dimensional q-state Potts model for q ⩽ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one-and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.     

Fractal structure of zeros in hierarchical models

We consider an Ising and aq-state Potts model on a diamond hierarchical lattice. We give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq. These zeros are just the Julia set corresponding to the renormalization group transformation.     

Mean-Field Criticality for Percolation on Planar Non-Amenable Graphs

The critical exponents β, γ, δ and Δ are proved to exist and to take their mean-field values for independent percolation on the following classes of infinite, locally finite, connected transitive graphs: (1) Non-amenable planar with one end. (2) Unimodular with infinitely many ends. Received: 4 April 2001 / Accepted: 4 October 2001