{"title":"低维范围自排斥弹性流形","authors":"C. Mueller, E. Neuman","doi":"10.31390/josa.3.2.01","DOIUrl":null,"url":null,"abstract":"We consider self-repelling elastic manifolds with a domain $[-N,N]^d \\cap \\mathbb{Z}^d$, that take values in $\\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for the case where $d \\geq 3$ and $D<d$ give a lower bound on $R_N$ of order $N^{\\frac{1}{D} \\left(d-\\frac{2(d-D)}{D+2} \\right)}$ and an upper bound proportional to $N^{\\frac{d}{2}+\\frac{d-D}{D+2}}$. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where $d=D$, which was studied by the authors in [10].","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"10 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Self-Repelling Elastic Manifolds with Low Dimensional Range\",\"authors\":\"C. Mueller, E. Neuman\",\"doi\":\"10.31390/josa.3.2.01\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider self-repelling elastic manifolds with a domain $[-N,N]^d \\\\cap \\\\mathbb{Z}^d$, that take values in $\\\\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for the case where $d \\\\geq 3$ and $D<d$ give a lower bound on $R_N$ of order $N^{\\\\frac{1}{D} \\\\left(d-\\\\frac{2(d-D)}{D+2} \\\\right)}$ and an upper bound proportional to $N^{\\\\frac{d}{2}+\\\\frac{d-D}{D+2}}$. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where $d=D$, which was studied by the authors in [10].\",\"PeriodicalId\":263604,\"journal\":{\"name\":\"Journal of Stochastic Analysis\",\"volume\":\"10 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Stochastic Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31390/josa.3.2.01\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Stochastic Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31390/josa.3.2.01","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Self-Repelling Elastic Manifolds with Low Dimensional Range
We consider self-repelling elastic manifolds with a domain $[-N,N]^d \cap \mathbb{Z}^d$, that take values in $\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for the case where $d \geq 3$ and $D
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