Extremes of the internal energy of the Potts model on cubic graphs. Issue 1 (4th February 2018)

• Record Type: Journal Article
• Authors: Davies, Ewan; Jenssen, Matthew; Perkins, Will; Roberts, Barnaby
• Is Part Of: Random structures & algorithms. Volume 53:Issue 1(2018)
• Page Start: 59
• Page End: 75
• Publication Date: 2018-02-04
• Subjects: graph homomorphims -- graph colorings -- Ising model -- partition function -- Potts model; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20767
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic graphs at every temperature and for all q ≥ 2 . This immediately implies corresponding tight bounds on the anti‐ferromagnetic Potts partition function. Taking the zero‐temperature limit gives new results in extremal combinatorics: the number of q ‐colorings of a 3‐regular graph, for any q ≥ 2, is maximized by a union of K 3, 3 's. This proves the d = 3 case of a conjecture of Galvin and Tetali.