Extremes of the internal energy of the Potts model on cubic graphs

Davies, Ewan; Jenssen, Matthew; Perkins, Will; Roberts, Barnaby

doi:10.1002/rsa.20767

Cited by 14 publications

(32 citation statements)

References 27 publications

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“…Since G × K 2 is bipartite, it follows by Theorem 1.11 that every such H satisfies Conjecture 1.15 (see Section 3.4). An example of such H is given by the adjacency matrix , and for these H, the conjecture has been verified for 3-regular [16] and 4-regular [14] graphs G via the occupancy method with computer assistance.…”

Section: Graphical Brascamp-lieb Inequalitiesmentioning

confidence: 94%

A reverse Sidorenko inequality

et al. 2020

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Let H be a graph allowing loops as well as vertex-and edge-weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms hom(G, H) satisfies the inequalitywhere du denotes the degree of vertex u in G. In particular, one has hom(G, H) 1/|E(G)| ≤ hom (K d,d , H) 1/d 2 for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of G is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings.For graph colorings, corresponding to H = Kq (also valid if some of the vertices of Kq are looped), we show that the triangle-free hypothesis on G may be dropped. A corollary is that among d-regular graphs, G = K d,d maximizes the quantity cq(G) 1/|V (G)| for every q and d, where cq(G) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, thenThis implies that among d-regular graphs, G = K d+1 maximizes hom(G, H) 1/|V (G)| . For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikvári-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn. arXiv:1809.09462v2 [math.CO] , where denotes a disjoint union. Question 1.1 was initially raised by Granville in 1988 in connection with the Cameron-Erdős conjecture on the number of sum-free sets. Alon [1] and Kahn [27] conjectured that G = K d,d is the exact maximizer. Alon [1] proved an asymptotic version as d → ∞, Kahn [27] proved the exact version under the additional hypothesis that G is bipartite, and Zhao [38] later removed this bipartite assumption. The results of Kahn [27] and Zhao [38] together answer Question 1.1: the maximizer is K d,d (unique up to taking disjoint unions of copies of itself). Galvin and Tetali [22] initiated the study of Questions 1.2 and 1.3 and extended Kahn's entropy method [27] to prove that, under the additional hypothesis that G is bipartite, G = K d,d is also the maximizer for hom(G, H) 1/|V (G)| . See Lubetzky and Zhao [32, Section 6] for a different proof using Hölder/Brascamp-Lieb type inequalities. Can the bipartite hypothesis on G also be dropped in this case? Not for all H: e.g., for H = , G = K d+1 is the maximizer instead of K d,d . Extending the technique for independent sets, Zhao [39] showed that the bipartite hypothesis can be dropped for certain classes of H, but the techniques failed for H = K q , corresponding to colorings (Question 1.2). It remained a tantalizing conjecture to remove the bipartite hypothesis for colorings.Recently, Davies, Jenssen, Perki...

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Section: Graphical Brascamp-lieb Inequalitiesmentioning

confidence: 94%

A reverse Sidorenko inequality

et al. 2020

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“…Conjecture 2.7 about the number of colorings was recently proved [22] for 3-regular graphs using an extension of the above method. Instead of the independence polynomial and the hard-core model, one considers a continuous relaxation of proper colorings by using the Potts model.…”

Section: Conjecture 73 ([10]mentioning

confidence: 98%

“…graph homomorphism coloring of surprising new consequences [24,22,48]. The results have been partially extended to graph homomorphisms, though many intriguing open problems remain.…”

Section: Theorem 12 ([55])mentioning

confidence: 99%

“…The conjecture was recently solved for d = 3 by Davies, Jenssen, Perkins, and Roberts [22] using a novel method they developed earlier. We will discuss the method in Section 7.…”

Section: Theorem 26 ([56]mentioning

confidence: 99%

“…Conjecture 7.4 was proved for 3-regular graphs in [22] using a variant of the method discussed in this section, by considering all configurations of the 2-step neighborhood of a uniform random vertex. The analysis is substantially more involved than the proofs we saw for independent sets.…”

Section: Consequently (By Integrating β and Notingmentioning

confidence: 99%

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