On a Factorization Formula for the Partition Function of Directed Polymers

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Tobias Hurth, Konstantin Khanin, Beatriz Navarro Lameda, Fedor Nazarov
{"title":"On a Factorization Formula for the Partition Function of Directed Polymers","authors":"Tobias Hurth,&nbsp;Konstantin Khanin,&nbsp;Beatriz Navarro Lameda,&nbsp;Fedor Nazarov","doi":"10.1007/s10955-023-03172-w","DOIUrl":null,"url":null,"abstract":"<div><p>We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice <span>\\(\\mathbb {Z}^{d+1}\\)</span>. The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, <i>t</i>], the error term is small uniformly over starting points <i>x</i> and endpoints <i>y</i> in the sub-ballistic regime <span>\\(\\Vert x - y \\Vert \\le t^{\\sigma }\\)</span>, where <span>\\(\\sigma &lt; 1\\)</span> can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime <span>\\(\\Vert x - y \\Vert \\le t^{1/2}\\)</span>. We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.</p></div>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":"190 10","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10589201/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10955-023-03172-w","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0

Abstract

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice \(\mathbb {Z}^{d+1}\). The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, t], the error term is small uniformly over starting points x and endpoints y in the sub-ballistic regime \(\Vert x - y \Vert \le t^{\sigma }\), where \(\sigma < 1\) can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime \(\Vert x - y \Vert \le t^{1/2}\). We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.

关于定向聚合物分配函数的因子分解公式。
我们证明了与时空晶格Zd+1上的定向聚合物模型相关的点对点配分函数的因子分解公式。聚合物受到由独立的同分布随机变量诱导的随机势的影响,我们考虑了弱无序状态,其中聚合物表现为扩散行为。我们证明,当将点对点分配函数的商和潜在随机游动的转移概率写成两个点对线分配函数加上一个误差项的乘积时,那么,对于大的时间间隔[0,t],在亚弹道状态中,误差项在起点x和终点y上均匀地小,其中σ1可以任意地接近1。这扩展了Sinai的一个结果,他证明了扩散区中误差项的小性。我们还导出了极限配分函数域的空间和时间相关性的渐近性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Journal of Statistical Physics
Journal of Statistical Physics 物理-物理:数学物理
CiteScore
3.10
自引率
12.50%
发文量
152
审稿时长
3-6 weeks
期刊介绍: The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.
×
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