[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85741-en":3,"doc-seo-85741-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85741,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Mixing and cutoff for the systematic scan dynamics of the mean-field ferromagnetic Potts model","Mixing time for the systematic scan dynamics is analyzed for the q-state ferromagnetic Potts model on the n-vertex complete graph (mean-field model). The study contrasts deterministic, fixed-order sequential vertex updates with Glauber dynamics, which updates a uniformly random vertex. For every q≥2 and inverse temperature β below the Glauber metastability threshold βs, the systematic scan chain mixes in Θ(log n) scans, i.e., Θ(n log n) single-site updates. A cutoff is proved: total variation drops abruptly within a narrow Θ(1) window, and results are tight as β exceeds βs.","arXiv :2607 .09841v1 [math .PR] 10 Jul 2026  \nMixing and cutoff for the systematic scan dynamics of the mean-field ferromagnetic Potts model  \nAntonio Blanca∗ Rafid Md Tahmidur†  \nAbstract  \nWe study the mixing time of the systematic scan dynamics for the q-state ferromagnetic Potts model on the n-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updatesa uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics.  \nWe take a step toward addressing this imbalance by showing that for every q ≥ 2 and β \u003C βs , where βs is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in Θ(log n) scans or, equivalently, in Θ(nlog n) single site updates. We in fact prove a sharper result; namely, that there exists a constant c (β, q) > 0 such that the mixing time is c(β, q)log n + Θ(1), which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow Θ(1) time window. This result is tight in β as well since the dynamics mixes exponentially slowly for β > βs. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.  \n∗Department of CSE, Pennsylvania State University, [ablanca@cse.psu.edu. Research supported in part by NSF CAREER grant](ablanca@cse.psu.edu. Research supported in part by NSF CAREER grant)[ ](ablanca@cse.psu.edu. Research supported in part by NSF CAREER grant)CCF-2143762 .  \n†Department of CSE, Pennsylvania State University, [mxr5997@psu.edu. Research](mxr5997@psu.edu. Research) supported in part by NSF CAREER grant CCF-2143762  \n1 Introduction  \nSampling high-dimensional probability distributions is an important task in science and engineering. It is, for example, a prevalent problem when running simulations in statistical physics or when solving inference problems in machine learning. As such, there is an array of sophisticated algorithms and heuristics to generate samples from these distributions. One of the most classical and powerful approaches is Gibbs Sampling, which iteratively updates one variable at a time conditioned on the values of all other variables.  \nThere are two standard strategies for deciding which variable to update in each step: (i) the Glauber dynamics, where the variable to be updated is chosen uniformly at random, and (ii) systematic scan dynamics, where variables are updated according to a predetermined ordering. In practice, systematic scan dynamics offer several computational advantages over the Glauber dynamics, as they are often amenable to parallelization, can exhibit improved memory locality and thus reduce data-access overhead, and require fewer random bits [28] . Despite these practical advantages, most theoretical analyses focus on the Glauber dynamics, which have historically proven more mathematically tractable, leaving the convergence theory for systematic scan dynamics significantly less developed.  \nThe number of variable updates required to reach stationarity can differ significantly between the Glauber and systematic scan dynamics, sometimes up to polynomial factors in n [20, 26] . For example, even in the trivial case of n independent variables, the systematic scan mixes in exactly n updates, whereas the Glauber dynamics requires ~~1~~2n log n updat","cbCaiiHymCRHXyi3","https://ap.wps.com/l/cbCaiiHymCRHXyi3","pdf",494495,1,40,"English","en",105,"# Abstract\n# Introduction\n## Sampling and Markov chain update rules\n## Mean-field ferromagnetic Potts model setup\n## Systematic scan dynamics definition and goal (mixing time)","[{\"question\":\"What dynamics are compared in this work?\",\"answer\":\"The paper compares systematic scan dynamics, which updates vertices sequentially in a fixed predetermined order, with Glauber dynamics, which updates a uniformly random vertex at each step.\"},{\"question\":\"Under what conditions does the systematic scan dynamics mix fast?\",\"answer\":\"For every q≥2 and inverse temperature β below the Glauber metastability threshold βs, the systematic scan dynamics mixes in Θ(log n) scans.\"},{\"question\":\"What cutoff result is established?\",\"answer\":\"The authors prove that the Markov chain exhibits the cutoff phenomenon: the total variation distance drops abruptly from nearly 1 to nearly 0 within a narrow Θ(1) time window.\"}]",1784205952,101,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"mixing-and-cutoff-for-the-systematic-scan-dynamics-of-the-mean-field-ferromagnetic-potts-model","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/mixing-and-cutoff-for-the-systematic-scan-dynamics-of-the-mean-field-ferromagnetic-potts-model/85741/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What dynamics are compared in this work?","Question",{"text":75,"@type":76},"The paper compares systematic scan dynamics, which updates vertices sequentially in a fixed predetermined order, with Glauber dynamics, which updates a uniformly random vertex at each step.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Under what conditions does the systematic scan dynamics mix fast?",{"text":80,"@type":76},"For every q≥2 and inverse temperature β below the Glauber metastability threshold βs, the systematic scan dynamics mixes in Θ(log n) scans.",{"name":82,"@type":73,"acceptedAnswer":83},"What cutoff result is established?",{"text":84,"@type":76},"The authors prove that the Markov chain exhibits the cutoff phenomenon: the total variation distance drops abruptly from nearly 1 to nearly 0 within a narrow Θ(1) time 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