Relations between scaling exponents in unimodular random graphs

Abstract

AbstractWe investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: $$\begin{aligned} d_{w} &= d_{f} + \tilde{\zeta }, \\ d_{s} &= 2 d_{f}/d_{w}, \end{aligned}$$ 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.