Periodic Ground States for the Potts Model with External Field and a Countable Set of Spin Values on the Cayley Tree

Mathematical Notes - For the Potts model with external field and a countable set of spin values on the Cayley tree of order $$k\ge2$$ , the set of periodic ground states corresponding to normal...     

Translation Invariance of the Periodic Gibbs Measures for the Potts Model on the Cayley Tree

Theoretical and Mathematical Physics - We study the Potts model with a zero external field on the Cayley tree. For the antiferromagnetic Potts model with q states on a second-order Cayley tree and...     

Periodic Gibbs measures for the Potts model on the Cayley tree

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures.     

Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree

We study translation-invariant Gibbs measures for the ferromagnetic Potts model with q states on the Cayley tree of order k and generalize some earlier results. We consider the question of the extremality of the known translation-invariant Gibbs measures for the Potts model with three states on the Cayley tree of order k = 3.     

The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree

We study the q-state Potts model on a Cayley tree of order k ≥ 2. In the group representation of the Cayley tree for the ferromagnetic Potts model, we single out a set of index-2 subgroups under which each weakly periodic Gibbs measure is translation invariant. For the anti-ferromagnetic Potts model with k ≥ 2 and q ≥ 2, we show that a weakly periodic Gibbs measure that is not translation invariant is not unique.     

The asymmetric random cluster model and comparison of Ising and Potts models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β _c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model. Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001     

The Tutte–Potts connection in the presence of an external magnetic field

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Noble and Welshʼs W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion–contraction reduction and Fortuin–Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.     

Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree

We study translation-invariant Gibbs measures on a Cayley tree of order k = 3 for the ferromagnetic three-state Potts model. We obtain explicit formulas for translation-invariant Gibbs measures. We also consider periodic Gibbs measures on a Cayley tree of order k for the antiferromagnetic q-state Potts model. Moreover, we improve previously obtained results: we find the exact number of periodic Gibbs measures with the period two on a Cayley tree of order k ≥ 3 that are defined on some invariant sets.     

Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

Mason’s Conjecture asserts that for an m-element rank r matroid the sequence is logarithmically concave, in which I _k is the number of independent k-sets of . A related conjecture in probability theory implies these inequalities provided that the set of independent sets of satisfies a strong negative correlation property we call the Rayleigh condition. This condition is known to hold for the set of bases of a regular matroid. We show that if ω is a weight function on a set system that satisfies the Rayleigh condition then is a convex delta-matroid and ω is logarithmically submodular. Thus, the hypothesis of the probabilistic conjecture leads inevitably to matroid theory. We also show that two-sums of matroids preserve the Rayleigh condition in four distinct senses, and hence that the Potts model of an iterated two-sums of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included. Research supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392.     

Comparison of Swendsen‐Wang and heat‐bath dynamics

We prove that the spectral gap of the Swendsen‐Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single‐spin dynamics. This implies rapid mixing for the two‐dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen‐Wang algorithm for planar graphs and prove rapid mixing for the two‐dimensional Potts models at all non‐critical temperatures. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 520–535, 2013     

New classes of ground states for the Potts model with random competing interactions on a Cayley tree

We consider the Potts model with random competing interactions on a Cayley tree. We study the domain of ground states of this model.     

Localization of translation-invariant Gibbs measures for the Potts and “solid-on-solid” models on a Cayley tree

We study Potts and “solid-on-solid” models with q ≥ 2 states on the Cayley tree of order k ≥ 1. For any values of the parameter q in the Potts model and q ≤ 6 in the “solid-on-solid” model, we find sets containing all translation-invariant Gibbs measures.     

The Classification of Certain Four-Weight Spin Models

In this paper we show that if one of the matrices {Wi, 1 h i h 4} of a four-weight spin model (X, W1, W2, W3, W4; D) is equivalent to the matrix of a Potts model or a cyclic model as type II matrix and |X| S 5, then the spin model is gauge equivalent to a Potts model or a cyclic model up to simultaneous permutations on rows and columns. Using this fact and Nomura's result [12] we show that every four-weight spin model of size |X| = 5 is gauge equivalent to either a Potts model or a cyclic model up to simultaneous permutations on rows and columns.     

On the Constructive Description of Gibbs Measures for the Potts Model on a Cayley Tree

Ukrainian Mathematical Journal - We consider the Potts model on a Cayley tree and prove the existence of Gibbs measures constructed by the method proposed in [H. Akin, U. A. Rozikov, and S. Temir,...     

Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree

For the Potts model with competing interactions, we describe the set of weakly periodic ground states corresponding to index-two normal divisors of the Cayley tree group representation. We also study some weakly periodic Gibbs measures.     

Ground states for the Potts model with competing interactions and a countable set of spin values on a Cayley tree

Theoretical and Mathematical Physics - We construct some finitely periodic ground states for the Potts model with competing interactions and a countable set of spin values on the Cayley tree of...     

The p-adic Potts model on the Cayley tree of order three

We study a phase transition problem for the q-state p-adic Potts model on the Cayley tree of order three. We find certain conditions for the existence of p-adic Gibbs measures and then establish the existence of a phase transition.     

Free energies of the Potts model on a Cayley tree

For the Potts model on the Cayley tree, we obtain some explicit formulas for the free energies and entropies in the case of vector-valued boundary conditions. These formulas include translation-invariant, periodic, and Dobrushin-like boundary conditions and also those corresponding to weakly periodic Gibbs measures.     

On Potts Model Clustering,Kernel K-Means and Density Estimation

Many clustering methods, such as K -means, kernel K -means, and MNcut clustering, follow the same recipe: (i) choose a measure of similarity between observations; (ii) define a figure of merit assigning a large value to partitions of the data that put similar observations in the same cluster; and (iii) optimize this figure of merit over partitions. Potts model clustering represents an interesting variation on this recipe. Blatt, Wiseman, and Domany defined a new figure of merit for partitions that is formally similar to the Hamiltonian of the Potts model for ferromagnetism, extensively studied in statistical physics. For each temperature T, the Hamiltonian defines a distribution assigning a probability to each possible configuration of the physical system or, in the language of clustering, to each partition. Instead of searching for a single partition optimizing the Hamiltonian, they sampled a large number of partitions from this distribution for a range of temperatures. They proposed a heuristic for choosing an appropriate temperature and from the sample of partitions associated with this chosen temperature, they then derived what we call a consensus clustering: two observations are put in the same consensus cluster if they belong to the same cluster in the majority of the random partitions. In a sense, the consensus clustering is an “average” of plausible configurations, and we would expect it to be more stable (over different samples)than the configuration optimizing the Hamiltonian.

The goal of this article is to contribute to the understanding of Potts model clustering and to propose extensions and improvements: (1) We show that the Hamiltonian used in Potts model clustering is closely related to the kernel K -means and MNCutcriteria. (2) We propose a modification of the Hamiltonian penalizing unequal clustersizes and show that it can be interpreted as a weighted version of the kernel K -meanscriterion. (3) We introduce a new version of the Wolff algorithm to simulate configurations from the distribution defined by the penalized Hamiltonian, leading to penalized Potts model clustering. (4) We note a link between kernel based clustering methods and nonparametric density estimation and exploit it to automatically determine locally adaptive kernel bandwidths. (5) We propose a new simple rule for selecting a good temperature T.

As an illustration we apply Potts model clustering to gene expression data and compare our results to those obtained by model based clustering and a nonparametric dendrogram sharpening method.