{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T22:50:42Z","timestamp":1786143042487,"version":"3.56.0"},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the performance of Glauber dynamics for the random cluster model with real parameter <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline1.png\"\/><jats:tex-math>\n$q\\gt 1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and temperature <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline2.png\"\/><jats:tex-math>\n$\\beta \\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Recent work by Helmuth, Jenssen, and Perkins detailed the ordered\/disordered transition of the model on random <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline3.png\"\/><jats:tex-math>\n$\\Delta$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-regular graphs for all sufficiently large <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline4.png\"\/><jats:tex-math>\n$q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and obtained an efficient sampling algorithm for all temperatures <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline5.png\"\/><jats:tex-math>\n$\\beta$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered\/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline6.png\"\/><jats:tex-math>\n$q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (with respect to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline7.png\"\/><jats:tex-math>\n$\\Delta$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover <jats:italic>all<\/jats:italic> temperatures <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000403_inline8.png\"\/><jats:tex-math>\n$\\beta$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered\/disordered transition and show spatial mixing properties \u2018within the phase\u2019, which are then related to the evolution of the chain.<\/jats:p>","DOI":"10.1017\/s0963548324000403","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T11:20:40Z","timestamp":1733743240000},"page":"359-391","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics"],"prefix":"10.1017","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3579-6531","authenticated-orcid":false,"given":"Andreas","family":"Galanis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1879-6089","authenticated-orcid":false,"given":"Leslie Ann","family":"Goldberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-8892-5170","authenticated-orcid":false,"given":"Paulina","family":"Smolarova","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"S0963548324000403_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01247839"},{"key":"S0963548324000403_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-023-04644-6"},{"key":"S0963548324000403_ref21","first-page":"817","article-title":"Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs","volume":"59","author":"Helmuth","year":"2023","journal-title":"Annales de l\u2019Institut Henri Poincare (B) Probabilites et statistiques"},{"key":"S0963548324000403_ref25","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOP737"},{"key":"S0963548324000403_ref16","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611977554.ch174"},{"key":"S0963548324000403_ref7","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS54457.2022.00040"},{"key":"S0963548324000403_ref2","first-page":"1","volume-title":"Approximation, Randomization, and Combinatorial Optimization. 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