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. Furthermore, under a weak form of spectral concentration, the spectral dimension coincides with the extremal volume growth exponent (this is the minimal exponent of volume growth under stationary changes of the graph metric).
Copyright. All Rights Reserved.