On the gauge Potts model and the plaquette percolation problem

It is shown that the ar 1, 0 limit of the Potts gauge model describes plaquette percolation as the analogous limit of the spin model describes bond percolation. These results further strengthen the connection between gauge theories and random surfaces. Moreover, further generalizations to other types of gauge theories are presented.     

Bond percolation on a finite lattice: The one-state Potts model reconsidered

Bond percolation on a finite lattice is studied by looking at the Kac mean field model. The investigation utilizes the one-state Potts model connection established by Kasteleyn and Fortuin. To deal with special problems associated with the finite extent of the system we re-cast the partition function, which allows us to investigate the percolation transition in detail. This fundamental new formulation clears up certain ambiguities present in previous treatments and indicates a possible new direction in the study of other replica-type models.     

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.     

Semi-infinite Ising model

For the semi-infinite Ising model in two or more dimensions, we prove analyticity properties of the surface free energy and map out the phase diagram in the absence of an external magnetic field. We prove that this phase diagram contains critical lines where the parallel and/or the transverse correlation lengths diverge. The critical exponent,v _⊥, of the transverse correlation length is shown to be equal to the exponentv of the Ising model on an infinite lattice. In a second paper, these results will be used to analyze the wetting transition.     

Semi-infinite Ising model

We study the equilibrium statistical mechanics of the semi-infinite Ising model, interpreted as a model of a binary system near a wall. In particular, the wetting transition is analyzed. In dimensionsd3 and at low temperature, we prove the existence of a layering transition which is of first-order.Dedicated to Walter Thirring on his 60^th birthday     

Pseudo-Hamiltonians for the Potts model at the critical point

The diagonal transfer matrix of the 2-dimensional, q-component Potts model on a square lattice is shown to commute with a linear operator at the critical point. In the 4-component model the linear operator is equivalent to the linear Heisenberg chain.     

Percolation and the Potts model

The Kasteleyn-Fortuin formulation of bond percolation as a lattice statistical model is rederived using an alternate approach. It is shown that the quantities of interest arising in the percolation problem, including the critical exponents, can be obtained from the solution of the Potts model. We also establish the Griffith inequality for critical exponents for the bond percolation problem.Work supported in part by NSF Grant No. D MR 76-20643.     

Potts model and graph theory

Elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the Potts model. Topics discussed include graphical expansions of the Potts partition function and correlation functions and their relationships with the chromatic, dichromatic, and flow polynomials occurring in graph theory. It is also shown that the Potts model realization of these classic graph-theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory.     

Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

Definition of percolation thresholds on self-affine surfaces

We study the percolation transition on a two-dimensional substrate with long-range self-affine correlations. We find that the position of the percolation threshold on a correlated lattice is no longer unique and depends on the spanning rule employed. Numerical results are provided for spanning across the lattice in specified (horizontal or vertical), either or both directions.     

Critical surfaces for general bond percolation problems

We present a general method for predicting bond percolation thresholds and critical surfaces for a broad class of two-dimensional periodic lattices, reproducing many known exact results and providing excellent approximations for several unsolved lattices. For the checkerboard and inhomogeneous bow-tie lattices, the method yields predictions that agree with numerical measurements to more than six figures, and are possibly exact.     

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     

Interface sharpness in the Potts model

A simple proof is given for the existence of a sharp interface between two ordered phases for the three-dimensional 2 ⁿ -state Potts model (n integer).     

Yang-Lee zeros of the Potts model

The Yang-Lee zeros of the three-component ferromagnetic Potts model in one dimension in the complex plane of an applied field are determined. The phase diagram consists of a triple point where three phases coexist. Emerging from the triple point are three lines on which two phases coexist and which terminate at critical points (Yang-Lee edge singularity). The zeros do not all lie on the imaginary axis but along the three two-phase lines. The model can be generalized to give rise to a tricritical point which is a new type of Yang-Lee edge singularity. Gibbs phase rule is generalized to apply to coexisting phases in the complex plane.Supported in part by the National Science Foundation under Grant No. DMR-81-06151.     

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.     

Paddle-wheel versus percolation model

Additional experimental results and interpretations are presented on the paddle-wheel versus percolation model for cation transport in Li₂SO₄-based compounds. These facts emphasize that the paddle-wheel model lacks concrete experimental support and that the proposed percolation model taken from percolation theory in interpreting the abrupt jump to fast ion conductivity in rotor and nonrotor solids undergoing a phase transition, i.e. structural percolation threshold, is far more appealing with a higher measure of validity.