Potts Automata

Abstract

The Potts model was introduced in Statistical Physics as a generalization of spin glasses [PoV,Pt,Wu]. Roughly described, the model comprises lattices where the spins may take several orientations rather than just two, as in the binary case (up and down). The Hamiltonian is:

$$ H\left( x \right) = - \frac{1}{2}\sum\limits_{{\left( {i,j} \right) \in v}} {{{\delta }_{K}}} \left( {{{x}_{i}},{{x}_{j}}} \right);\quad {{x}_{j}} \in Q = \left\{ {0, \ldots ,q - 1} \right\} $$

(7.1)

where Q is the finite set of orientations, (i, j)∈ V means the sites i and j are neighbours, and δ_K is the Kroeneker function: δ_K(u,v) = 1 iff u = v. It is not difficult to see that (7.1) reduces to the Ising Hamiltonian (up to an additive constant) for |Q| = 2. In this chapter we suppose that the lattice interactions occur in a non-oriented graph G = (I, V), where I = {1,....,n} is the set of sites, while the set of links V is assumed to be symmetric without loops: (i, j) ∉ V.