Title: Conformal growth rates and spectral geometry on distributional limits of graphs,
Abstract: 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 {\em 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 the almost
sure spectral dimension.
In two dimensions, one obtains more precise information. If $(G,\rho)$ has a
property we call {\em quadratic conformal growth}, then the following holds: If
the degree of the root is uniformly bounded almost surely, then $G$ is almost
surely recurrent. Since limits of finite $H$-minor-free graphs have gauged
quadratic conformal growth, such limits are almost surely recurrent; this
affirms a conjecture of Benjamini and Schramm (2001). For the special case of
planar graphs, this gives a proof of the Benjamini-Schramm Recurrence Theorem
that does not proceed via the analysis of circle packings.
Gurel-Gurevich and Nachmias (2013) resolved a central open problem by showing
that the uniform infinite planar triangulation (UIPT) and quadrangulation
(UIPQ) are almost surely recurrent. They proved that this holds for any
distributional limit of planar graphs in which the degree of the root has
exponential tails (which is known to hold for UIPT and UIPQ). We use the
quadratic conformal growth property to give a new proof of this result that
holds for distributional limits of finite $H$-minor-free graphs. Moreover, our
arguments yield quantitative bounds on the heat kernel in terms of the degree
distribution at the root. This also yields a new approach to subdiffusivity of
the random walk on UIPT/UIPQ, using only the volume growth profile of balls in
the intrinsic metric.