In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.

We present exact calculations of flow polynomials F(G,q) for lattice strips of various fixed widths Ly and arbitrarily great lengths Lx, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G,q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros B in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case Ly = 1, the maximal point, qcf, where B crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value qcf = 3 for the infinite square lattice.

We consider the problem of estimating the partition function of the ferromagnetic q-state Potts model. We propose an importance sampling algorithm in the dual of the normal factor graph representing the model. The algorithm can efficiently compute an estimate of the partition function when the coupling parameters of the model are strong (corresponding to models at low temperature) or when the model contains a mixture of strong and weak couplings. We show that, in this setting, the proposed algorithm significantly outperforms the state of the art methods.

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We will try to sketch Professor F. Y. Wu’s contributions in lattice statistical mechanics, solid state physics, graph theory, enumerative combinatorics and so many other domains of physics and mathematics. We will recall F. Y. Wu’s most important and well-known classic results, and we will also sketch his most recent research dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics. Since it is hard to provide an exhaustive list of all his contributions, to give some representation of F. Y. Wu’s “mental connectivity”, we will concentrate on the interrelations between the various results he has obtained in so many different domains of physics and mathematics. Along the way we will also try to understand Wu’s motivations and his favorite concepts, tools and ideas. PACS. 05.50.+q – Lattice theory and statistics; Ising problems. I.

We will try to sketch Professor F. Y. Wu’s contributions in lattice statistical mechanics, solid state physics, graph theory, enumerative combinatorics and so many other domains of physics and mathematics. We will recall F. Y. Wu’s most important and well-known classic results and we will also sketch his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics. Since it is hard to provide an exhaustive list of all his contributions, to give some representation of F. Y. Wu’s ”mental connectivity” we will concentrate on the interrelations between the various results he has obtained in so many different domains of physics and mathematics. Along the way we will also try to understand Wu’s motivations and his favorite concepts, tools and ideas.

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