Relations between scaling exponents in unimodular random graphs

Abstract

We 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 d_w is the walk dimension, d_f is the fractal dimension, d_s 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 d_f and \(\tilde{\zeta } \geqslant 0\) exist, then d_w and d_s 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 d_w=4 using d_f=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 d_w=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that d_w=d_f for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since d_f>2.