  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.
  The consequences in dimension two are particularly strong. It establishes
that models like the uniform infinite planar triangulation (UIPT) and
quadrangulation (UIPQ) almost surely have spectral dimension at most two. It
also establishes a conjecture of Benjamini and Schramm (2001) by extending
their Recurrence Theorem from planar graphs to arbitrary families of $H$-minor
free graphs. More generally, it strengthens the work of Gurel-Gurevich and
Nachmias (2013) who established recurrence for distributional limits of planar
graphs when the degree of the root has exponential tails.
  We further present a general method for proving subdiffusivity of the random
walk on a large class of models, including UIPT and UIPQ, using only the volume
growth profile of balls in the intrinsic metric.
