{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,8]],"date-time":"2025-05-08T12:43:16Z","timestamp":1746708196168},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T00:00:00Z","timestamp":1662508800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2023,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline1.png\" \/><jats:tex-math>\n$4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-state anti-ferromagnetic Potts model with interaction parameter <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline2.png\" \/><jats:tex-math>\n$w\\in (0,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on the infinite <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline3.png\" \/><jats:tex-math>\n$(d+1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-regular tree has a unique Gibbs measure if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline4.png\" \/><jats:tex-math>\n$w\\geq 1-\\dfrac{4}{d+1_{_{\\;}}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline5.png\" \/><jats:tex-math>\n$d\\geq 4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This is tight since it is known that there are multiple Gibbs measures when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline6.png\" \/><jats:tex-math>\n$0\\leq w\\lt 1-\\dfrac{4}{d+1}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline7.png\" \/><jats:tex-math>\n$d\\geq 4$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We moreover give a new proof of the uniqueness of the Gibbs measure for the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline8.png\" \/><jats:tex-math>\n$3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-state Potts model on the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline9.png\" \/><jats:tex-math>\n$(d+1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-regular tree for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline10.png\" \/><jats:tex-math>\n$w\\geq 1-\\dfrac{3}{d+1}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline11.png\" \/><jats:tex-math>\n$d\\geq 3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline12.png\" \/><jats:tex-math>\n$w\\in (0,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000207_inline13.png\" \/><jats:tex-math>\n$d=2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548322000207","type":"journal-article","created":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T04:46:21Z","timestamp":1662525981000},"page":"158-182","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Uniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular tree"],"prefix":"10.1017","volume":"32","author":[{"given":"David","family":"de Boer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pjotr","family":"Buys","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guus","family":"Regts","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,9,7]]},"reference":[{"key":"S0963548322000207_ref7","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1899"},{"key":"S0963548322000207_ref4","doi-asserted-by":"publisher","DOI":"10.1137\/18M1219722"},{"key":"S0963548322000207_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51829-9"},{"key":"S0963548322000207_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-014-0947-5"},{"key":"S0963548322000207_ref12","doi-asserted-by":"publisher","DOI":"10.1145\/2785964"},{"key":"S0963548322000207_ref8","first-page":"247","article-title":"Random colorings of a Cayley tree","volume":"10","author":"Brightwell","year":"2002","journal-title":"Contemp. Comb."},{"key":"S0963548322000207_ref3","doi-asserted-by":"publisher","DOI":"10.4171\/AIHPD\/108"},{"key":"S0963548322000207_ref5","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1935"},{"key":"S0963548322000207_ref16","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/16\/4\/018"},{"key":"S0963548322000207_ref21","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2010.34"},{"key":"S0963548322000207_ref22","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOP888"},{"key":"S0963548322000207_ref19","doi-asserted-by":"publisher","DOI":"10.1137\/16M1101003"},{"key":"S0963548322000207_ref24","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132538"},{"key":"S0963548322000207_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548315000401"},{"key":"S0963548322000207_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(87)90267-6"},{"key":"S0963548322000207_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100027419"},{"key":"S0963548322000207_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/9781316882603"},{"key":"S0963548322000207_ref2","unstructured":"[2] Bencs, F. , de Boer, D. , Buys, P. and Regts, G. (2022) Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite \u0394-regular tree for large \u0394. arXiv preprint arXiv: 2203.15457."},{"key":"S0963548322000207_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441"},{"key":"S0963548322000207_ref9","unstructured":"[9] Efthymiou, C. (2020) On sampling symmetric Gibbs distributions on sparse random graphs and hypergraphs. arXiv preprint arXiv: 2007.07145."},{"key":"S0963548322000207_ref11","first-page":"43","article-title":"Uniqueness for the 3-state antiferromagnetic Potts model on the tree","volume":"23","author":"Galanis","year":"2018","journal-title":"Paper No., Electron. J. 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Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}