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

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.

💡 Research Summary

The paper investigates the computational complexity of approximating the partition function of the ferromagnetic q‑state Potts model on bounded‑degree graphs, focusing on bipartite graphs of maximum degree Δ. Building on the earlier result of Goldberg and Jerrum, which showed that approximating the ferromagnetic Potts partition function is #BIS‑hard in general, the authors refine this hardness to the setting of bipartite graphs with a degree bound.

The first technical contribution is a precise analysis of the phase diagram of the model on the infinite Δ‑regular tree T_Δ. Unlike the Ising model, the ferromagnetic Potts model exhibits a first‑order (discontinuous) phase transition: for a critical interaction strength B₀ (given explicitly in equation (3)), the system switches from a unique Gibbs measure (the disordered phase where all colors appear equally) to a regime where q symmetric ordered phases dominate, each favoring one color with marginal probability a = a(q, B, Δ). The authors characterize B₀ by studying the tree recursion (belief propagation) and identifying the attractive fixed points that correspond to stable phases.

Armed with this thermodynamic picture, the authors prove that for any B > B₀ the problem #BipFerroPotts(q, B, Δ) – computing the partition function of the q‑state ferromagnetic Potts model on a bipartite graph of maximum degree Δ – is #BIS‑hard under approximation‑preserving reductions. The reduction uses random bipartite Δ‑regular graphs as gadgets. They introduce a “first‑moment functional” Ψ₁(α) = lim_{n→∞} (1/n) log E