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The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety of objects defined on graphs. This chapter shows that they permit a representation of the two-point correlation function of a ferromagnetic Potts model on a graph G in terms of the flow polynomials of certain related random graphs. This representation extends to general Potts models the so-called random-current expansion for Ising models, and it amplifies the links between the Potts partition function and the Tutte polynomial.

This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are used to derive the first 25 virial coefficients of the virial expansion. The analyticity of the pressure in the density plane in both the low and high density regions are then presented. The general theory of the chiral Potts model as a two-dimensional statistical model is presented and the eigenvalues of the three-component superintegrable case are computed in detail. The order parameter is discussed and the phase diagram of the general three-component integrable chiral Potts spin chain is given. Open questions are discussed about Q operators, eight-vertex model degeneracies, and conjectures for correlations functions of the superintegrable chiral Potts model.

This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.

Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.

This chapter illustrates basic concepts of quantum integrable systems on two important models of statistical physics: the Q-state Potts model and the O(n) model. Both models are transformed into loop and vertex models that provide representations of the dense and dilute Temperley–Lieb algebras. The identification of the corresponding integrable R-matrices leads to the solution of both models by the algebraic Bethe Ansatz technique. Elementary excitations are discussed in the critical case and the link to conformal field theory in the thermodynamic limit is established. The concluding sections outline the solution of a specific model of the theta point of collapsing polymers, leading to a continuum limit with a non-compact target space.

Chapter 2 discusses one-dimensional statistical models, for example, the Ising model and its generalizations (Potts model, systems with O(n) or Zn-symmetry, etc.). It discusses several methods of solution and covers the recursive method, the transfer matrix approach, and series expansion techniques. General properties of these methods, which are valid on higher-dimensional lattices, are also covered. The contents of this chapter are quite simple and pedagogical but extremely useful for understanding the following sections of the book. One of the appendices at the end of the chapter is devoted to a famous problem of topology, i.e. the four-colour problem, and its relation with the two-dimensional Potts model.

Chapter 14 discusses how the identification of a class of universality is one of the central questions needing an answer for those in the field of statistical physics. This chapter discusses in detail the class of universality of several models, and provides examples that include the Ising model, the tricritical Ising model and its structure constants, the Yang–Lee model and the 3-state Potts model. This chapter also covers the study of the statistical models of geometric type (as, for instance, those that describe the self-avoiding walks) and their formulation in terms of conformal minimal models, including conformal models with O(n) symmetry.