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: $d_w = d_f + \tilde{\zeta}$ and $d_s = 2 d_f/d_w$, 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 $\tilde{\zeta} \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned equalities hold. Moreover, our primary new estimate is the relation $d_w \geq d_f + \tilde{\zeta}$, which 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 2003) and $\tilde{\zeta}=0$ (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2017 and Ding-Gwynne 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 $d_f > 2$ (Ding and Gwynne 2020). For the random walk on $\mathbb{Z}^2$ driven by conductances from an exponentiated Gaussian free field with exponent $\gamma > 0$, one has $d_f = d_f(\gamma)$ and $\tilde{\zeta}=0$ (Biskup, Ding, and Goswami 2020). This yields $d_s=2$ and $d_w = d_f$, confirming two predictions of those authors.