Scaling exponents in stationary random graphs

Stationary random graphs provide a rich family of random geometries for studying conjectured relationships between scaling exponents that arise in the statistical physics literature. Here we examine the relationships between the fractal dimension, the walk dimension, the resistance exponent, the spectral dimension, and the extremal growth exponent.

In the recurrent regime, we show that the "Einstein relations" hold in generality, so that the density and conductivity of a stationary random graph determine the rate of escape of the random walk and the spectral dimension. We further show that, under a weak form of spectral concentration, the spectral dimension coincides with the extremal volume growth exponent (the minimal exponent of volume growth under stationary reweightings of the graph metric).