  Consider an infinite planar graph with uniform polynomial growth of degree d
> 2. Many examples of such graphs exhibit similar geometric and spectral
properties, and it has been conjectured that this is necessary. We present a
family of counterexamples. In particular, we show that for every rational d >
2, there is a planar graph with uniform polynomial growth of degree d on which
the random walk is transient, disproving a conjecture of Benjamini (2011).
  By a well-known theorem of Benjamini and Schramm, such a graph cannot be a
unimodular random graph. We also give examples of unimodular random planar
graphs of uniform polynomial growth with unexpected properties. For instance,
graphs of (almost sure) uniform polynomial growth of every rational degree d >
2 for which the speed exponent of the walk is larger than 1/d, and in which the
complements of all balls are connected. This resolves negatively two questions
of Benjamini and Papasoglou (2011).
