{"title":"Spatial mixing and the random‐cluster dynamics on lattices","authors":"Reza Gheissari, Alistair Sinclair","doi":"10.1002/rsa.21191","DOIUrl":null,"url":null,"abstract":"Abstract An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"17 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21191","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.