[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81681-en":3,"doc-seo-81681-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81681,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Counting and Sampling Anti-Ferromagnetic Potts Models on Random Regular Bipartite Graphs in the Non-Uniqueness Regime","Investigates low-temperature approximation and sampling for the anti-ferromagnetic q-state Potts model on random regular bipartite graphs, which are likely to be good bipartite expanders. Builds on the uniqueness/non-uniqueness phase transition picture and links it to Markov-chain mixing and partition-function approximation complexity. Shows exponentially slow mixing of single-site Glauber dynamics when temperature is sufficiently low relative to graph degree, and provides a deterministic partition-function approximation via abstract polymer models.","arXiv :2606 .2 1250v2 [ cs .DS] 10 Jul 2026  \nCOUNTING AND SAMPLING ANTI-FERROMAGNETIC POTTS MODELSON RANDOM REGULAR BIPARTITE GRAPHS IN THE NON-UNIQUENESS  \nREGIME  \nZHIDAN LI, SIYU LIU, KUAN YANG  \nAbstract. The anti-ferromagnetic multi-state Potts model, a generalization of the Ising model, isone of the most fundamental models in statistical physics. It was conjectured by Koteck´y (Phys. Rev. B, 1985) that the model undergoes a phase transition from a disordered phase at infinite temperature to an ordered phase at sufficiently low temperature on lattices. Such phase transitions are believed to play an important role in computational complexity theory and remain closely connected to the problem of approximating the partition function of the system. For proper three-coloring models (corresponding to the zero-temperature), torpid mixing of a family of local-update Markov chains on lattices was established by Galvin, Kahn, Randall and Sorkin (SIDMA, 2015), coinciding with the presence of phase coexistence following shown by Feldheim and Spinka (J. Eur. Math. Soc., 2019) . In this work, we study approximating the partition function of the anti-ferromagnetic multi-state  \nPotts model at low temperature on random regular bipartite graphs, which are with high probability good bipartite expanders. On the negative side, we generalize the result by Geisler, Kang, Sarantisand Wdowinski (arXiv, 2026) for anti-ferromagnetic Ising models to show that when the temperature is sufficiently low relative to the degree of the underlying graph, the celebrated single-site Glauber dynamics has exponentially slow mixing time. On the positive side, we design a deterministic algorithm that yields an approximation to the partition function of the model via the framework of abstract polymer models as Jenssen, Keevash and Perkins (SICOMP, 2020), Liao, Lin, Lu and Mao (Theor. Comput. Sci. , 2022), Galanis, Goldberg and Stewart (TOCT, 2021) and Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026) .  \n(Zhidan Li) School of Computer Science, Shanghai Jiao Tong University, Shanghai, China. Email: [yueEnTeRnAL@sjtu.edu.cn](yueEnTeRnAL@sjtu.edu.cn).  \n(Siyu Liu) Zhiyuan College, Shanghai Jiao Tong University, Shanghai, China. Email: williamliusy@ [sjtu.edu.cn](sjtu.edu.cn).  \n(Kuan Yang) John Hopcroft Center for Computer Science, Shanghai Jiao Tong University, Shanghai, China. Email: [kuan.yang@sjtu.edu.cn](kuan.yang@sjtu.edu.cn).  \n1. Introduction  \nThe q-state Potts model, introduced by R. B. Potts [Pot52], is a fundamental spin system in statistical physics and has been well studied in probability theory and theoretical computer science. It generalizes the Ising model by allowing each site to take one of q possible colors. Formally, given a ground set U, the model is specified by a color set [q] := {1,..., q} for a positive integer q ≥ 3, an inverse temperature β and a Hamiltonian H : [q]U → R. A configuration is an assignment σ : U → [q] and the weight of σ is defined by:  \n(1) w (σ) := e−βH(σ) .  \nLet the state space or configuration space Ω := [q]U be the collection of all configurations. When β \u003C 0, we say that it is a ferromagnetic q-Potts model and an anti-ferromagnetic case for β > 0.  \nWe study the anti-ferromagnetic q-Potts model on graphs. Namely, for a graph G = (V, E), the ground set is V (and thus Ω = [q]V ), and the Hamiltonian H (·) is given by the number of monochromatic edges in G under a configuration, i. e. ,  \n(2) ∀σ ∈ Ω, H(σ) = mG (σ) := |{ (u, v) ∈ E : σ|u = σ|v}| .  \nThe Gibbs distribution induced by the anti-ferromagnetic q-Potts model on G at β is the probability distribution µ = µG,q;β on Ω defined as  \n w (σ)   e−βmG (σ)  \n∀σ ∈ Ω, µ(σ) = =  \nZG,q(β) ZG,q(β)  \nwhere ZG,q(β) = P σ∈Ω w (σ) is the partition function. In particular, when the temperature is zero (β = ∞ ), configurations with monochromatic edges receive weight 0 and configurations representing proper colorings receive weight 1 . Thus the proper vertex-coloring model is a ","cbCaivI6aVDiiiHp","https://ap.wps.com/l/cbCaivI6aVDiiiHp","pdf",446504,4,1,21,"English","en",105,"# Introduction\n## Anti-ferromagnetic Potts model and partition function\n## Approximate counting and complexity (FPTAS/FPRAS)\n## Uniqueness threshold on infinite regular trees","[{\"question\":\"What is the main computational problem studied for the anti-ferromagnetic Potts model?\",\"answer\":\"The central problem is approximating the partition function ZG,q(β), since exact computation is #P-complete even on restricted graph classes.\"},{\"question\":\"What result is obtained for sampling via Glauber dynamics at low temperature?\",\"answer\":\"When the temperature is sufficiently low compared with the underlying graph degree, single-site Glauber dynamics has exponentially slow mixing time.\"},{\"question\":\"How is the partition function approximated on random regular bipartite graphs?\",\"answer\":\"A deterministic approximation algorithm is designed using the framework of abstract polymer models to approximate the partition function at low temperature.\"}]",1784175384,53,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"counting-and-sampling-anti-ferromagnetic-potts-models-on-random-regular-bipartite-graphs-in-the-non-uniqueness-regime","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/counting-and-sampling-anti-ferromagnetic-potts-models-on-random-regular-bipartite-graphs-in-the-non-uniqueness-regime/81681/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 is the main computational problem studied for the anti-ferromagnetic Potts model?","Question",{"text":75,"@type":76},"The central problem is approximating the partition function ZG,q(β), since exact computation is #P-complete even on restricted graph classes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What result is obtained for sampling via Glauber dynamics at low temperature?",{"text":80,"@type":76},"When the temperature is sufficiently low compared with the underlying graph degree, single-site Glauber dynamics has exponentially slow mixing time.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the partition function approximated on random regular bipartite graphs?",{"text":84,"@type":76},"A deterministic approximation algorithm is designed using the framework of abstract polymer models to approximate the partition function at low 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