{"pk":64951,"title":"A near-optimal zero-free disk for the Ising model","subtitle":null,"abstract":"The partition function of the Ising model of a graph \\(G=(V,E)\\) is defined as \\(Z_{\\operatorname{Ising}}(G;b)=\\sum_{\\sigma:V\\to \\{0,1\\}} b^{m(\\sigma)}\\), where \\(m(\\sigma)\\) denotes the number of edges \\(e=\\{u,v\\}\\) such that \\(\\sigma(u)=\\sigma(v)\\). We show that for any positive integer \\(\\Delta\\) and any graph \\(G\\) of maximum degree at most \\(\\Delta\\), \\(Z_{\\operatorname{Ising}}(G;b)\\neq 0\\) for all \\(b\\in \\mathbb{C}\\) satisfying \\(|\\frac{b-1}{b+1}| \\leq \\frac{1-o_\\Delta(1)}{\\Delta-1}\\) (where \\(o_\\Delta(1) \\to 0\\) as \\(\\Delta\\to \\infty\\)). This is optimal in the sense that \\(\\tfrac{1-o_\\Delta(1)}{\\Delta-1}\\) cannot be replaced by \\(\\tfrac{c}{\\Delta-1}\\) for any constant \\(c › 1\\) subject to a complexity theoretic assumption.\nTo prove our result we use a standard reformulation of the partition function of the Ising model as the generating function of even sets. We establish a zero-free disk for this generating function inspired by techniques from statistical physics on partition functions of polymer models. Our approach is quite general and we discuss extensions of it to certain types of polymer models.\n \nMathematics Subject Classifications: 05C31, 82B20, 68W25\n \nKeywords: Ising model, partition function, even set, polymer model, Fisher zeros, approximate counting","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Ising model"},{"word":"partition function"},{"word":"even set"},{"word":"polymer model"},{"word":"Fisher zeros"},{"word":"approximate counting"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/2q5853zz","frozenauthors":[{"first_name":"Viresh","middle_name":"","last_name":"Patel","name_suffix":"","institution":"School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, U.K.","department":""},{"first_name":"Guus","middle_name":"","last_name":"Regts","name_suffix":"","institution":"Korteweg de Vries Institute for Mathematics, University of Amsterdam. P.O. Box 94248, 1090 GE, Amsterdam, The Netherlands","department":""},{"first_name":"Ayla","middle_name":"","last_name":"Stam","name_suffix":"","institution":"Cygnus Gymnasium. Vrolikstraat 8, 1091 VG, Amsterdam, The Netherlands","department":""}],"date_submitted":"2024-09-26T14:14:56Z","date_accepted":"2024-09-26T14:14:56Z","date_published":"2024-09-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64951/galley/49761/download/"}]}