[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84650-en":3,"doc-seo-84650-105":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},84650,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Mixing Times of Spin Systems on Dynamical Percolation","Mixing times are studied for stochastic nearest-neighbour spin systems evolving with Glauber dynamics on a dynamical percolation environment. Spins live on a d-dimensional torus of side length N, while each edge independently alternates open/closed at rate λ using Ber(p). When an edge is open, spin updates occur at rate 1 according to Gibbs-type conditional measures restricted to open edges. For a broad class of systems with p \u003C pc(d), and any temperature, sufficiently small λ yields mixing time of order log(λN). The Markov chain is non-reversible, and the argument uses a coupling of local configurations when the environment behaves well.","arXiv :2607 .02477v1 [math .PR] 2 Jul 2026  \nMixing times of spin systems on dynamical  \npercolation  \nAlexandre Stauffer∗ Oskar Vavtar†  \nJuly 3, 2026  \nAbstract  \nWe study the mixing times of stochastic spin systems corresponding to nearestneighbour Glauber dynamics on dynamical percolation, defined on d-dimensional torus of side-length N. In this model, the status of each edge (open or closed) updates independently at rate λ > 0, according to Ber(p) samples. Simultaneously, the spin of each site updates at rate 1 according to Glauber dynamics on the environment restricted to open edges. We show that for a relatively general class of nearest-neighbour systems, as long as p \u003C pc(d), for any temperature, if λ is sufficiently small, the mixing time is of order ~~log~~λN . This Markov chain is non-reversible, and the proof is obtained by developing a particular coupling that couples together local configurations whenever the environment behaves well.  \n1 Introduction  \nLet G = (V, E) be a finite graph and S a finite set. We study the dynamics of a class of SV-valued spin systems in a random dynamic environment. The latter will be given by dynamical percolation, which is a {0 , 1}E-valued Markov chain, which represents opening and closing of edges in E.  \nFixing p ∈ (0 , 1) and λ > 0, we define the dynamical percolation with speed λ and density p, which we denote by (ηt)t≥0, as follows:  \n(i) we start from an arbitrary (possibly random) initial configuration η0 ∈ {0 , 1}E ;  \n(ii) each edge carries an independent rate-λ Poisson clock;  \n(iii) if the clock corresponding to edge e rings at time t, we resample the value of ηt (e) according to Ber(p), independently of anything else.  \n∗ Department of Mathematics, King’s College London. Email: [a.stauffer@kcl.ac.uk](a.stauffer@kcl.ac.uk).†Department of Mathematics, King’s College London. Email: [oskar.vavtar@kcl.ac.uk](oskar.vavtar@kcl.ac.uk).  \nBy convention, we say that the edge e is open (resp. closed) at time t if ηt (e) = 1 (resp. ηt (e) = 0) . It is easy to check that (ηt)t≥0 is invariant with respect to Bernoulli (bond) percolation on G, i.e., the measure Pp = PGp, given by  \nPp (η) = Y pη (e)(1 − p)1−η(e), η ∈ {0 , 1}E .  \ne∈E  \nConsider now a collection of potentials Φ = {Φe : e ∈ E}, where for each e ∈ E , the map Φe : S × S → R is symmetric. That is, for e = xy and σ ∈ SV , we have  \nΦxy (σ(x),σ (y)) = Φxy (σ(y),σ (x)) .  \nWe define a spin system given by potentials Φ on dynamical percolation as a stochastic process (σt ,ηt)t≥0, which evolves as follows:  \n(i) (ηt)t≥0 is taken to be dynamical percolation on {0 , 1}E with speed λ and density p.  \n(ii) (σt)t≥0 is the SV-valued spin component, started from arbitrary σ0 ∈ SV , which evolves in the following way:  \n(a) each site carries an independent rate-1 Poisson clock;  \n(b) if the clock corresponding to site x rings at time t, the value of σt (x) resamples according to  \nµxσt,ηt (·) :=  exp 􀀀−Py:xy∈E ηt (xy)Φxy (·,σt (y)) 􀀁  (1 . 1)  \nPs∈S exp 􀀀−Py:xy∈E ηt (xy)Φxy (s,σt (y)) 􀀁 .  \nWe will refer to this process as Φ-spin system on dynamical percolation. The process (σt ,ηt)t≥0 is also a Markov chain; assuming that Φ is such that the process is irreducible, we will write π for the corresponding invariant measure. Note, however, that (σt)t≥0 by itself is not a Markov chain, as the transition rates depend on the environment. Moreover, the chain (σt ,ηt)t≥0 is not reversible, in the sense that there is no measure on SV × {0 , 1}E with respect to which the chain is reversible.  \nWe note that if in the part (i) of the definition above, we did not consider dynamical percolation but simply took ηt ≡ 1 for all t ≥ 0, we would recover the Glauber dynamics for the Gibbs measure given by the Hamiltonian H =Pxy∈E Φxy . In this case we could without loss of generality simply consider only the first coordinate (σt)t≥0 which would indeed be a Markov chain. Throughout the paper, we will refer to this process as the Φ -Glauber dynamics on G. ","cbCailOpkeACMl5Q","https://ap.wps.com/l/cbCailOpkeACMl5Q","pdf",563800,2,1,37,"English","en",105,"# Introduction\n## Dynamical percolation environment\n## Φ-spin systems and Glauber dynamics\n## Mixing time definition and main result","[{\"question\":\"What stochastic dynamics couple spins and the percolation environment?\",\"answer\":\"Edges evolve as dynamical percolation with density p, independently resampled at rate λ. Spins update at rate 1 using Glauber dynamics conditioned on which edges are open, based on the given interaction potentials Φ.\"},{\"question\":\"What conditions on p and λ ensure fast mixing?\",\"answer\":\"For nearest-neighbour systems in a general class, if p \\u003c pc(d) and λ is sufficiently small (for any temperature), the mixing time is of order log(λN).\"},{\"question\":\"Why is the resulting Markov chain non-reversible, and how does the proof proceed?\",\"answer\":\"The joint process (σt, ηt) is stated to be non-reversible, with no measure making it reversible on SV × {0,1}E. The proof uses a specific coupling that synchronizes local configurations when the environment behaves well.\"}]",1784197481,93,{"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},"mixing-times-of-spin-systems-on-dynamical-percolation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/mixing-times-of-spin-systems-on-dynamical-percolation/84650/",4,{"url":51,"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","2026-07-22","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 stochastic dynamics couple spins and the percolation environment?","Question",{"text":75,"@type":76},"Edges evolve as dynamical percolation with density p, independently resampled at rate λ. Spins update at rate 1 using Glauber dynamics conditioned on which edges are open, based on the given interaction potentials Φ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What conditions on p and λ ensure fast mixing?",{"text":80,"@type":76},"For nearest-neighbour systems in a general class, if p \u003C pc(d) and λ is sufficiently small (for any temperature), the mixing time is of order log(λN).",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the resulting Markov chain non-reversible, and how does the proof proceed?",{"text":84,"@type":76},"The joint process (σt, ηt) is stated to be non-reversible, with no measure making it reversible on SV × {0,1}E. The proof uses a specific coupling that synchronizes local configurations when the environment behaves well.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]