Moments of Partition Functions of 2d Gaussian Polymers in the Weak Disorder Regime-I
Moments of Partition Functions of 2d Gaussian Polymers in the Weak Disorder Regime-I
复制标题
弱无序域-I 中二维高斯聚合物配分函数的矩
DOI：
10.1007/s00220-023-04799-2
复制
发表时间：
2021
影响因子：
2.4
通讯作者：
O. Zeitouni
中科院分区：
文献类型：
--
作者：
Clément Cosco;O. Zeitouni
Let WN(β)=E0e∑n=1Nβω(n,Sn)-Nβ2/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_N(\beta ) = {\mathrm E}_0\left[ e^{ \sum _{n=1}^N \beta \omega (n,S_n) - N\beta ^2/2}\right] $$\end{document} be the partition function of a two-dimensional directed polymer in a random environment, where ω(i,x),i∈N,x∈Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (i,x), i\in \mathbb {N}, x\in \mathbb {Z}^2$$\end{document} are i.i.d. standard normal and {Sn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{S_n\}$$\end{document} is the path of a random walk. With β=βN=β^π/logN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =\beta _N={\hat{\beta }} \sqrt{\pi /\log N}$$\end{document} and β^∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}\in (0,1)$$\end{document} (the subcritical window), logWN(βN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log W_N(\beta _N)$$\end{document} is known to converge in distribution to a Gaussian law of mean -λ2/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\lambda ^2/2$$\end{document} and variance λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^2$$\end{document}, with λ2=log(1/(1-β^2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^2=\log (1/(1-{\hat{\beta }}^2))$$\end{document} (Caravenna et al. in Ann Appl Probab 27(5):3050–3112, 2017). We study in this paper the moments E[WN(βN)q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {E}}}[W_N( \beta _N)^q]$$\end{document} in the subcritical window, for q=O(logN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=O(\sqrt{\log N})$$\end{document}. The analysis is based on ruling out triple intersections.
Let WN(β)=E0e∑n=1Nβω(n,Sn)-Nβ2/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_N(\beta ) = {\mathrm E}_0\left[ e^{ \sum _{n=1}^N \beta \omega (n,S_n) - N\beta ^2/2}\right] $$\end{document} be the partition function of a two-dimensional directed polymer in a random environment, where ω(i,x),i∈N,x∈Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (i,x), i\in \mathbb {N}, x\in \mathbb {Z}^2$$\end{document} are i.i.d. standard normal and {Sn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{S_n\}$$\end{document} is the path of a random walk. With β=βN=β^π/logN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =\beta _N={\hat{\beta }} \sqrt{\pi /\log N}$$\end{document} and β^∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}\in (0,1)$$\end{document} (the subcritical window), logWN(βN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log W_N(\beta _N)$$\end{document} is known to converge in distribution to a Gaussian law of mean -λ2/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\lambda ^2/2$$\end{document} and variance λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^2$$\end{document}, with λ2=log(1/(1-β^2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^2=\log (1/(1-{\hat{\beta }}^2))$$\end{document} (Caravenna et al. in Ann Appl Probab 27(5):3050–3112, 2017). We study in this paper the moments E[WN(βN)q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {E}}}[W_N( \beta _N)^q]$$\end{document} in the subcritical window, for q=O(logN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=O(\sqrt{\log N})$$\end{document}. The analysis is based on ruling out triple intersections.
登录
查看更多内容
临界二维随机热流
影响因子：
3.1
作者：
Caravenna F
通讯作者：
Caravenna F
构造 $(2 1)$ 维 KPZ 方程的解
影响因子：
2.3
作者：
Chatterjee, Sourav;Dunlap, Alexander
通讯作者：
Dunlap, Alexander
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