{"title":"具有凸势的∇φ界面Gibbs采样器的谱隙和截止现象","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":"{\"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}","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}
Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ interfaces with convex potential
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.
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