Relations between scaling exponents in unimodular random graphs

Understand

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.