Scaling exponents in stationary random graphs
Many random networks arising in statistical physics are stochastically homogeneous and can be modeled as stationary random graphs. Often in these models, certain asymptotic geometric and spectral properties are known (or conjectured) to have scaling exponents. These include the fractal dimension (d_f), the walk dimension (d_w), the spectral dimension (d_s), and the resistance exponent (R).
By envisioning the graph as a homogeneous substrate with prescribed density and conductivity, the "Einstein relations" assert the plausible set of equalities d_w = d_f + R and d_s = 2 d_f/d_w. A long sequence of works has rigorously verified these equalities in the setting of "strongly recurrent" graphs. We establish these relations in the more general regime where R is nonnegative, using the additional feature of stochastic homogeneity (formally, unimodularity of the random graph model). This has a number of concrete applications to well-studied models like critical 2D percolation and the uniform infinite planar triangulation (UIPT).
Taking inspiration from geometric analysis on manifolds, we show that the spectral dimension can often be characterized as the infimal rate of volume growth over all (random) changes of metric on the graph. This provides a useful tool for confirming some long-standing experimental predictions from the statistical physics literature.