[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83146-en":3,"doc-seo-83146-105":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},83146,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Mixing of Glauber Dynamics on High Overlap Gibbs Measures","The paper establishes fast mixing of Glauber dynamics for specific quadratic Gibbs measures on the binary hypercube under large external fields. The core mechanism is a high-overlap condition that uniformly controls correlation matrices across all pinnings, achieved by bounding norms of small submatrices of the interaction matrix. With stochastic localization, the work derives a lower bound on the spectral gap and hence polynomial-time mixing. It applies these results to the Sherrington–Kirkpatrick model, using a scaled GOE interaction matrix.","arXiv :2607 .06813v1 [math .PR] 7 Jul 2026  \nMixing of Glauber Dynamics on High Overlap Gibbs Measures  \nAfonso S. Bandeira∗ Ahmed El Alaoui† Almut Rödder ‡  \nJuly 9, 2026  \nAbstract  \nWe show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics.  \nAs a direct application, we consider the Sherrington–Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature β , there exists a strength of external field θ, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix) .  \nContents  \n1 Introduction 1  \n1. 1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.2 Proof Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.3 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2 High Overlap Distributions 5  \n2.1 Correlation Bounds based on High Overlap ......................... 5  \n2.2 Spin Alignment in Quadratic Gibbs Measures ........................ 7  \n3 Proof of the Main Results 9  \n3.1 Stochastic Localization with Pinnings ............................ 9  \n3.2 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10  \n3.3 Proof of Corollary 1.2 ..................................... 11  \n4 Further Applications and Open Questions 12  \n4.1 Z2-Synchronization with side information .......................... 12  \n1 Introduction  \nThe main object of study of this paper are quadratic Gibbs measures over the binary hypercube {±1}n  \nν (x) ∝ exp 􀀒 12x⊤ Jx + h⊤ x􀀓 (1)  \n∗ ASB: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, [Switzerland.](Switzerland. bandeira@math.ethz.ch)[ bandeira@math.ethz.ch](Switzerland. bandeira@math.ethz.ch)[ ](Switzerland. bandeira@math.ethz.ch)†AE: Department of Statistics and Data Science, Cornell University, Ithaca, New York, [USA.](USA. elalaoui@cornell.edu)[ elalaoui@cornell.edu](USA. elalaoui@cornell.edu)[ ](USA. elalaoui@cornell.edu)‡AR: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland.  \n[almutmagdalena.roedder@ifor.math.ethz.ch](almutmagdalena.roedder@ifor.math.ethz.ch)  \nwhere J ∈ Rn ×n is a symmetric interaction matrix and h ∈ Rn an external field. Probability distributions of the form (1) are a central object of study in several fields, including Statistical Physics where different choices of J and h capture models, such as Curie-Weiss, Ising, Sherrington-Kirkpatrick, Edwards-Anderson, among others. In Statistics, such distributions arise as posterior distributions of various statistical models such as Z2 Synchronization and Community Detection in the Stochastic Block Model.  \nOne of the most important questions surrounding these probability distributions is whether they can be sampled efficiently, meaning in time polynomial in n. A particularly important sampling algorithm is Glauber Dynamics, a Markov Chain Monte Carlo algorithm where, at each iteration, a random spin gets picked and its spin redrawn at random with respect to its conditional distribution (also referred to as a heat bath on one coordinate) .  \nThere is a rich line of work showing fast mixing of Glauber Dynamics under various conditions on J (e.g. ([EKZ22, CE25 , AKV24]) . Particularly remarkable is the spectral radius condition of Eldan, Koehler, and Zeitouni which guarantees fast mixing of Glauber from any starting point as long as the spectral diameter of J is","cbCaihpp9SOSS9v4","https://ap.wps.com/l/cbCaihpp9SOSS9v4","pdf",525530,2,1,15,"English","en",105,"# Introduction\n## Main Results\n## Proof Strategy\n## Notation and Definitions\n# High Overlap Distributions\n## Correlation Bounds based on High Overlap\n## Spin Alignment in Quadratic Gibbs Measures\n# Proof of the Main Results\n## Stochastic Localization with Pinnings\n## Proof of Theorem 1.1\n## Proof of Corollary 1.2\n# Further Applications and Open Questions\n## Z2-Synchronization with side information","[{\"question\":\"What is the main result about Glauber dynamics in this paper?\",\"answer\":\"The paper proves fast mixing of Glauber dynamics for certain quadratic Gibbs measures when the external field is large, by obtaining a spectral gap lower bound.\"},{\"question\":\"How does the paper control correlations when proving mixing?\",\"answer\":\"It uses a high-overlap condition to uniformly bound correlation matrices over all pinnings, by controlling norms of small submatrices of the interaction matrix.\"},{\"question\":\"What application is provided for the Sherrington–Kirkpatrick model?\",\"answer\":\"For the Sherrington–Kirkpatrick model with a scaled GOE interaction matrix, it shows that for any fixed finite inverse temperature β there exists an external field strength θ (independent of system size) such that Glauber dynamics mixes in polynomial time with high probability over the draw of the interaction matrix.\"}]",1784185597,38,{"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-of-glauber-dynamics-on-high-overlap-gibbs-measures","",{"@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-of-glauber-dynamics-on-high-overlap-gibbs-measures/83146/",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 is the main result about Glauber dynamics in this paper?","Question",{"text":75,"@type":76},"The paper proves fast mixing of Glauber dynamics for certain quadratic Gibbs measures when the external field is large, by obtaining a spectral gap lower bound.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper control correlations when proving mixing?",{"text":80,"@type":76},"It uses a high-overlap condition to uniformly bound correlation matrices over all pinnings, by controlling norms of small submatrices of the interaction matrix.",{"name":82,"@type":73,"acceptedAnswer":83},"What application is provided for the Sherrington–Kirkpatrick model?",{"text":84,"@type":76},"For the Sherrington–Kirkpatrick model with a scaled GOE interaction matrix, it shows that for any fixed finite inverse temperature β there exists an external field strength θ (independent of system size) such that Glauber dynamics mixes in polynomial time with high probability over the draw of the interaction matrix.","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 & 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