Abstract
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} $$\\begin{aligned} d_{w} &= d_{f} + \ ilde{\\zeta }, \\\\ d_{s} &= 2 d_{f}/d_{w}, \\end{aligned}$$ \\end{document} where dw is the walk dimension, df is the fractal dimension, ds is the spectral dimension, and tilde{zeta } 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 if df and tilde{zeta } geqslant 0 exist, then dw and ds exist, and the aforementioned equalities hold. Moreover, our primary new estimate d_{w} geqslant d_{f} + tilde{zeta } is established for all tilde{zeta } in mathbb{R}.For the uniform infinite planar triangulation (UIPT), this yields the consequence dw=4 using df=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and tilde{zeta }=0 (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, 2020)). The conclusion dw=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that dw=df for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since df>2.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have