Computing the Potts Free Energy and Submodular Functions

A growing part of statistical mechanics relies now on the use of simulation and computers. New exact results are extremely scarce, and it appears that almost all exact results attainable with known methods have been found. This is particularly true with disordered models. In this context numerical methods become very important. Various generalizations of the Ising model have been introduced. Two types of reasoning motivated these generalizations – some were intended to describe more realistic situations, while the merit of the others was solvability. Exact solutions for disordered models exist mainly for one-dimensional models, or for infinite dimensional models (mean-field models). In two and three dimensions various approximate solutions exist, most of them relying on heavy numerical calculations. A variety of numerical methods have been designed to study these models. In particular, low-temperature behavior leads to combinatorial optimization problems, since in this limit only the states of minimum energy contribute. It is worthwhile noticing at this point that the exchanges between numerical statistical mechanics and combinatorial optimization have been useful in both directions: several improvements in the simulated annealing method are now in common use in optimization on one hand, while, conversely, properties of the ground states of the disordered Ising model have been elucidated using optimization methods. The problem we address in this chapter is not a low-temperature problem. On the contrary, we will compute the free energy of a Potts model at any temperature, including some phase transition temperatures. To transform the problem of computing the free energy into an optimization problem (i.e. to find a minimum in a finite set), we need to take some limit. Usually this is a zero-temperature limit. Here this will be the limit of an infinite number of states. At first sight this limit seems quite academic, since no experimental situation at this time is described by the infinite state Potts model. However, this model is still of importance since it is very likely that it belongs to the same universality class as the Random Transverse Quantum Ising chain, a model which has a very interesting behavior under renormalization. This chapter is organized as follows: in the first part the Potts model is introduced. In the second part the submodular functions are introduced. Only a very few results and examples are presented: from this very active field of discrete mathematics, only the strict minimum needed to follow the algorithm of the last section is covered. The references included are far from being exhaustive. In the third part the connection between submodular function and the