A little statistical mechanics for the graph theorist

Abstract

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.

Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

“…The chromatic polynomial is a special case of an important two-variable function, namely the partition function of the q-state Potts model [9] , Z(G, q, v) with v = −1 (zero-temperature Potts antiferromagnet), or equivalently, the Tutte polynomial T (G, x, y) [1, [10][11][12]] with x = 1 − q and y = 0 (see Eqs. (A6) and (A14) in the Appendix; some recent reviews are [13][14][15]). Using this connection, one can equivalently express a(G) as an evaluation of T (G, x, y), namely a(G) = T (G, 2, 0) .…”

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

“…The q-state Potts model has served as a valuable system for the study of phase transitions and critical phenomena [1]- [5], and recently there has been considerable interest in its connections with mathematical graph theory [6]- [8]. For two-dimensional lattices, additional insights into the critical behavior have been obtained from conformal algebra methods [9,10].…”

“…Transfer matrix and related linear algebraic methods, as well as related generating function methods, have also been used to calculate a particular special case in zero field, namely the chromatic polynomial [13,14,30]; references to the literature can be found in reviews such as Refs. [2]- [8].…”

“…Because (1.4) does not contain any explicit reference to the spins {σ i } or summation over spin configurations, it allows one to define the zero-field Potts model partition function with q not necessarily restricted to the positive integers, N + . The zero-field Potts model partition function is equivalent to the Tutte (also called Tutte-Whitney) polynomial T (G, x, y), a function of major importance in mathematical graph theory [6]- [8], [12]- [14] defined by 5) where c(G…”

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

“…Since then, it has become the darling of statistical mechanics, both for physicists and mathematicians [4,64], as one of few "exactly soluble" (or at least tractable) models demonstrating a phase transition [11,15,16,27,31,35]. Due to its intuitive appeal to describe multistate systems, combined with a rich structure of inner symmetries, the Potts model has been quickly picked up by a host of research in diverse areas, such as probability [25], algebra [33], graph theory [5], conformally invariant scaling limits [46,54], computer science [18], statistics [23,39], biology [24], medicine [58,59], sociology [53,55], financial engineering [45,60], computational algorithms [10,17], technological processes [52,62], and many more. Much of this modelling has involved interacting spin system on graphs.…”