$$\nabla \phi $$ 界面模型和缩放极限的 Ray-Knight 定理

IF 1.5 1区 数学 Q2 STATISTICS & PROBABILITY
Jean-Dominique Deuschel, Pierre-François Rodriguez
{"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}
引用次数: 0

摘要

我们引入了一种自然度量,它适用于在与凸势能梯度吉布斯度量相关的朗格文动力学驱动下的随时间变化的环境中演化的双无限随机漫步轨迹。我们推导出了该量度诱导的泊松云占据时间与相应梯度场平方之间的关系,而梯度场一般不是高斯的。在二次情况下,我们恢复了著名的第二雷-奈特定理的广义。我们进一步确定了维度 3 中涉及的各种对象的缩放极限,发现它们表现出均质化。特别是,我们证明了梯度场的重规范化平方在适当的重缩放下收敛于具有适当扩散矩阵的高斯自由场在 \(\mathbb R^3\) 上的威克有序平方,从而扩展了纳达夫和斯宾塞关于场本身的缩放极限的著名结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Probability Theory and Related Fields
Probability Theory and Related Fields 数学-统计学与概率论
CiteScore
3.70
自引率
5.00%
发文量
71
审稿时长
6-12 weeks
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信