{"title":"$$\\nabla \\phi $$ 界面模型和缩放极限的 Ray-Knight 定理","authors":"Jean-Dominique Deuschel, Pierre-François Rodriguez","doi":"10.1007/s00440-024-01275-3","DOIUrl":null,"url":null,"abstract":"<p>We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is <i>not</i> Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on <span>\\(\\mathbb R^3\\)</span> with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.\n</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"51 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Ray–Knight theorem for $$\\\\nabla \\\\phi $$ interface models and scaling limits\",\"authors\":\"Jean-Dominique Deuschel, Pierre-François Rodriguez\",\"doi\":\"10.1007/s00440-024-01275-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is <i>not</i> Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on <span>\\\\(\\\\mathbb R^3\\\\)</span> with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.\\n</p>\",\"PeriodicalId\":20527,\"journal\":{\"name\":\"Probability Theory and Related Fields\",\"volume\":\"51 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2024-04-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Probability Theory and Related Fields\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00440-024-01275-3\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Theory and Related Fields","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00440-024-01275-3","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
A Ray–Knight theorem for $$\nabla \phi $$ interface models and scaling limits
We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on \(\mathbb R^3\) with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.
期刊介绍:
Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.