Approximating the partition function of the ferromagnetic Potts model

TL;DR

This paper demonstrates the computational difficulty of approximating the partition function of the ferromagnetic q-state Potts model for q>2, linking it to the complexity of various counting problems via phase transition analysis.

Contribution

It proves the hardness of approximating the partition function of the ferromagnetic Potts model for q>2, using phase transition properties of the random cluster model.

Findings

Approximating the partition function is #RHPi_1-hard for q>2.

The problem is as hard as counting independent sets in bipartite graphs.

The proof uses the phase transition in the random cluster model.

Abstract

We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q-state Potts model when q>2. Specifically we show that the partition function is hard for the complexity class #RHPi_1 under approximation-preserving reducibility. Thus, it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems, including that of determining the number of independent sets in a bipartite graph. Our proof exploits the first order phase transition of the "random cluster" model, which is a probability distribution on graphs that is closely related to the q-state Potts model.