Relations between scaling exponents in unimodular random graphs
Relations between scaling exponents in unimodular random graphs
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DOI：
10.1007/s00039-023-00654-7
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发表时间：
2020-07
影响因子：
2.2
通讯作者：
James R. Lee
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文献类型：
--
作者：
James R. Lee
我们在单模随机网络的一般背景下研究了“爱因斯坦关系”的有效性。以下是与缩放指数相关的等式：\DocentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\Begin{aliged}d_{w}&=d_{f}+\tilde{\zeta}，\\d_{S}&=2d_{f}/d_{w}，其中是行走维度，df是分形维，d是谱维，和\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\tilde{\zeta}$\end{Document}是阻力指数。粗略地说，这与随机游走者的平均位移和返回概率与底层介质的密度和导电性有关。我们证明了ifdfand\Documentclass[12pt]{minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\tilde{Zeta}\geqslant 0$\end{Document}存在，并且上述等式成立。此外，我们的主要新预估\Documentclass[12pt]{Minimal}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$d_{w}\geqslant d_{f}+\tilde{\zeta}$\end{DocumentDocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\tilde{\zeta}\in\mathbb{R}$\end{Document}。对于统一无限平面三角剖分(UIPT)，这产生了结果w=4，使用df=4(Geom中的天使。功能。肛门。13(5)：935-974，)和\Documentclass[12pt]{Minimal}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\tilde{\Zeta}=0$\end{Document}(在此建立是基于Liouville量子引力理论，继Gwynne-Miller 2020和(Ding And Gwynne In Commun)之后。数学课。太棒了。374(3)：1877-1934，))。结论dw=4以前是Gwynne和Hutchcroft(2018)使用更精细的方法建立的。一个新的结果是一致无限Schnyder-wood装饰三角剖分的dw=df，这意味着简单的随机游动是次扩散的，因为f>；2.
We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} d_{w} &= d_{f} + \tilde{\zeta }, \\ d_{s} &= 2 d_{f}/d_{w}, \end{aligned}$$ \end{document} wheredwis the walk dimension,dfis the fractal dimension,dsis the spectral dimension, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{\zeta }$\end{document} is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that ifdfand \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{\zeta } \geqslant 0$\end{document} exist, thendwanddsexist, and the aforementioned equalities hold. Moreover, our primary new estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{w} \geqslant d_{f} + \tilde{\zeta }$\end{document} is established for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{\zeta } \in \mathbb{R}$\end{document}.For the uniform infinite planar triangulation (UIPT), this yields the consequencedw=4 usingdf=4 (Angel in Geom. Funct. Anal. 13(5):935–974, ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{\zeta }=0$\end{document} (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, )). The conclusiondw=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is thatdw=dffor the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, sincedf>2.