Relations between scaling exponents in unimodular random graphs

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.