The Scaling Limit of the Directed Polymer with Power-Law Tail Disorder
The Scaling Limit of the Directed Polymer with Power-Law Tail Disorder
复制标题
具有幂律尾部无序的定向聚合物的尺度极限
DOI：
10.1007/s00220-021-04082-2
复制
发表时间：
2020
影响因子：
2.4
通讯作者：
H. Lacoin
中科院分区：
文献类型：
--
作者：
Q. Berger;H. Lacoin
In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk (Sn)n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_n)_{n\ge 0}$$\end{document} on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document}, with d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 1$$\end{document}, and modify its law using Gibbs weights in the product form ∏n=1N(1+βηn,Sn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod _{n=1}^{N} (1+\beta \eta _{n,S_n})$$\end{document}, where (ηn,x)n≥0,x∈Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta _{n,x})_{n\ge 0, x\in {\mathbb {Z}}^d}$$\end{document} is a field of i.i.d. random variables whose distribution satisfies P(η>z)∼z-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}(\eta >z) \sim z^{-\alpha }$$\end{document} as z→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow \infty $$\end{document}, for some α∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document}. We prove that if α<min(1+2d,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < \min (1 + \frac{2}{d} ,2)$$\end{document}, when sending N to infinity and rescaling the disorder intensity by taking β=βN∼N-γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =\beta _N \sim N^{-\gamma }$$\end{document} with γ=d2α(1+2d-α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\frac{d}{2\alpha }(1+\frac{2}{d}-\alpha )$$\end{document}, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable noise constructed in the companion paper (Berger and Lacoin in The continuum directed polymer in Lévy Noise, 2020. arXiv:2007.06484v2).
In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk (Sn)n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_n)_{n\ge 0}$$\end{document} on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document}, with d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 1$$\end{document}, and modify its law using Gibbs weights in the product form ∏n=1N(1+βηn,Sn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod _{n=1}^{N} (1+\beta \eta _{n,S_n})$$\end{document}, where (ηn,x)n≥0,x∈Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta _{n,x})_{n\ge 0, x\in {\mathbb {Z}}^d}$$\end{document} is a field of i.i.d. random variables whose distribution satisfies P(η>z)∼z-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}(\eta >z) \sim z^{-\alpha }$$\end{document} as z→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow \infty $$\end{document}, for some α∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document}. We prove that if α<min(1+2d,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < \min (1 + \frac{2}{d} ,2)$$\end{document}, when sending N to infinity and rescaling the disorder intensity by taking β=βN∼N-γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =\beta _N \sim N^{-\gamma }$$\end{document} with γ=d2α(1+2d-α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\frac{d}{2\alpha }(1+\frac{2}{d}-\alpha )$$\end{document}, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable noise constructed in the companion paper (Berger and Lacoin in The continuum directed polymer in Lévy Noise, 2020. arXiv:2007.06484v2).
登录
查看更多内容
影响因子：
2.6
作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
通讯作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
DOI：
10.1214/19-aop1383
发表时间：
2018-12
期刊：
The Annals of Probability
影响因子：
--
作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
通讯作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
关于$$(2 1)$$-维定向聚合物的矩和临界窗口中的随机热方程
影响因子：
2.4
作者：
Caravenna F
通讯作者：
Caravenna F
二维 SHE 的关键时刻
DOI：
10.2140/pmp.2021.2.179
发表时间：
2021
期刊：
Probability and Mathematical Physics
影响因子：
--
作者：
Gu, Yu;Quastel, Jeremy;Tsai, Li-Cheng
通讯作者：
Tsai, Li-Cheng