Conformal growth rates and spectral geometry on distributional limits of graphs

Understand

For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices.

• This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting.
• Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension.
• This has interesting consequences for many low-dimensional models.