Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

TL;DR

This paper refines the understanding of computational hardness for the ferromagnetic Potts model, establishing #BIS-hardness for bipartite graphs of maximum degree D below a critical temperature, and analyzes phase transitions and algorithmic mixing times.

Contribution

It provides a detailed phase diagram for the infinite D-regular tree, refines #BIS-hardness results for bipartite graphs, and extends analytical tools to ferromagnetic models on random regular graphs.

Findings

Refined phase diagram and critical temperature for the infinite D-regular tree.

Proved #BIS-hardness for approximating the partition function below the critical temperature.

Showed torpid mixing of Swendsen-Wang algorithm at critical temperature for large q.

Abstract

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.