{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:37:21Z","timestamp":1759639041388},"reference-count":20,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":1492,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2006]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice <jats:bold>Z<\/jats:bold><jats:sup>2<\/jats:sup>. The model has two parameters: <jats:italic>q<\/jats:italic>, the number of spins, and \u03bb = exp(\u2212\u03b2), where \u03b2 4 is \u2018inverse temperature\u2019. It is known that the model has strong spatial mixing if <jats:italic>q<\/jats:italic> &gt; 7, or if <jats:italic>q<\/jats:italic> = 7 and \u03bb = 0 or \u03bb &gt; 1\/8, or if <jats:italic>q<\/jats:italic> = 6 and \u03bb = 0 or \u03bb &gt; 1\/4. The \u03bb = 0 case corresponds to the model in which configurations are proper <jats:italic>q<\/jats:italic>-colourings of <jats:bold>Z<\/jats:bold><jats:sup>2<\/jats:sup>. It is shown that the system has strong spatial mixing for <jats:italic>q<\/jats:italic> \u2265 6 and any \u03bb. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of \u03bb than was previously known for <jats:italic>q<\/jats:italic> = 3, <jats:italic>q<\/jats:italic> = 4 and <jats:italic>q<\/jats:italic> = 5.<\/jats:p>","DOI":"10.1112\/s1461157000001169","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T07:41:55Z","timestamp":1375774915000},"page":"1-20","source":"Crossref","is-referenced-by-count":10,"title":["Improved Mixing Bounds for the Anti-Ferromagnetic Potts Model on <b>Z<\/b><sup>2<\/sup>"],"prefix":"10.1112","volume":"9","author":[{"given":"Leslie Ann","family":"Goldberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Markus","family":"Jalsenius","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Russell","family":"Martin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mike","family":"Paterson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000001169_ref012","doi-asserted-by":"publisher","DOI":"10.1137\/0222066"},{"key":"S1461157000001169_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20004"},{"key":"S1461157000001169_ref010","unstructured":"10. Jalsenius M. , PhD thesis, University of Warwick, in preparation."},{"key":"S1461157000001169_ref019","unstructured":"19. Weitz D. , \u2018Mixing in time and space for discrete spin systems\u2019, PhD thesis, University of California, Berkeley, 2004."},{"key":"S1461157000001169_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070205"},{"key":"S1461157000001169_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF02101930"},{"key":"S1461157000001169_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20002"},{"key":"S1461157000001169_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096629"},{"key":"S1461157000001169_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10090"},{"key":"S1461157000001169_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF02199113"},{"key":"S1461157000001169_ref008","article-title":"Strong spatial mixing with fewer colours for lattice graphs","author":"Goldberg","journal-title":"SIAM J. 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