{"title":"Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ interfaces with convex potential","authors":"P. Caputo, Cyril Labbé, H. Lacoin","doi":"10.1214/21-aihp1174","DOIUrl":null,"url":null,"abstract":"We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on R describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gapN = 1 − cos(π/N), and that for all ǫ ∈ (0, 1), its ǫ-mixing time satisfies TN (ǫ) ∼ logN 2 gapN as N → ∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential. MSC 2010 subject classifications: Primary 60J25; Secondary 37A25, 82C22.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/21-aihp1174","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 9
Abstract
We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on R describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gapN = 1 − cos(π/N), and that for all ǫ ∈ (0, 1), its ǫ-mixing time satisfies TN (ǫ) ∼ logN 2 gapN as N → ∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential. MSC 2010 subject classifications: Primary 60J25; Secondary 37A25, 82C22.