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

Abstract

Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree ∆ undergoes a phase transition that coincides with the uniqueness/nonuniqueness phase transition on the infinite ∆-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, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree ∆. To this end, we first present a detailed picture for the phase diagram for the infinite ∆-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 (corresponding to the region where the ordered phase "dominates") that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree ∆.The #BIS-hardness result uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent results establishing connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. In this paper we extend these connections to random regular graphs for all ferromagnetic models. Using these connections, we establish the Bethe prediction for every ferromagnetic spin system on random regular graphs, which says roughly that the expectation of the log of the partition function Z is the same as the log of the expectation of Z. As a further consequence of our results, we prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing (i. e., exponentially slow convergence to its stationary distribution) on random ∆-regular graphs at the critical temperature for sufficiently large q.

Counting independent sets in unbalanced bipartite graphs

“…Now since Condition (9) holds with constant η > 0, we can use (12) and (15): To prove Theorem 1.3 we need to do two things. First we need to show that the exponential decay of joint cumulants in the minimum Steiner tree size implies that |µ A∪B − µ A µ B | decays exponentially in the distance from A to B; second we need to extend these results from sets of vertices contained solely in R to general sets of vertices in L ∪ R. Then…”

Swendsen‐Wang algorithm on the mean‐field Potts model

“…In this perspective, corresponds to the uniqueness/non‐uniqueness threshold on ; corresponds to the ordered/disordered phase transition; and was conjectured by Häggström to correspond to a second uniqueness/non‐uniqueness threshold for the random‐cluster model on with periodic boundaries (in particular, he conjectured that non‐uniqueness holds iff ). For a detailed exposition of these critical points we refer the reader to (see also for their relevance for random regular graphs). We should finally remark that in the case of the Ising model ( q = 2), the three points coincide.…”

“…In particular, Gore and Jerrum showed for the complete graph that the mixing time is at the critical point . Similar slow mixing results have been established for other classes of graphs at the analogous critical point: Cooper and Frieze showed this for G ( n , p ) when , Galanis et al for random Δ‐regular graphs when , and Borgs et al for the d ‐dimensional integer lattice for . For square boxes of , Ullrich proves polynomial mixing time at all temperatures except criticality building upon the results of Beffara and Duminil‐Copin .…”

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

“…Since then, it has become the darling of statistical mechanics, both for physicists and mathematicians [4,64], as one of few "exactly soluble" (or at least tractable) models demonstrating a phase transition [11,15,16,27,31,35]. Due to its intuitive appeal to describe multistate systems, combined with a rich structure of inner symmetries, the Potts model has been quickly picked up by a host of research in diverse areas, such as probability [25], algebra [33], graph theory [5], conformally invariant scaling limits [46,54], computer science [18], statistics [23,39], biology [24], medicine [58,59], sociology [53,55], financial engineering [45,60], computational algorithms [10,17], technological processes [52,62], and many more. Much of this modelling has involved interacting spin system on graphs.…”

“…In particular, regular trees (known as Cayley trees or Bethe lattices [6]) have become a standard trial template for various models of statistical physics (see, e.g., [1,2,3,34,37,61,63]), which are interesting in their own right but also provide useful insights into (harder) models in more realistic spaces (such as lattices Z d ) as their "infinite-dimensional" approximation [4,Chapter 4]. On the other hand, the use of Cayley trees is often motivated by the applications, such as information flows [38] and reconstruction algorithms on networks [15,36], DNA strands and Holliday junctions [48], evolution of genetic data and phylogenetics [14], bacterial growth and fire forest models [12], or computational complexity on graphs [18]. Crucially, the criticality in such models is governed by phase transitions in the underlying spin systems.…”