  We consider the effect of quenched spatial disorder on systems of
interacting, pinned non-Abelian anyons as might arise in disordered Hall
samples at filling fractions \nu=5/2 or \nu=12/5. In one spatial dimension,
such disordered anyon models have previously been shown to exhibit a hierarchy
of infinite randomness phases. Here, we address systems in two spatial
dimensions and report on the behavior of Ising and Fibonacci anyons under the
numerical strong-disorder renormalization group (SDRG). In order to manage the
topology-dependent interactions generated during the flow, we introduce a
planar approximation to the SDRG treatment. We characterize this planar
approximation by studying the flow of disordered hard-core bosons and the
transverse field Ising model, where it successfully reproduces the known
infinite randomness critical point with exponent \psi ~ 0.43. Our main
conclusion for disordered anyon models in two spatial dimensions is that
systems of Ising anyons as well as systems of Fibonacci anyons do not realize
infinite randomness phases, but flow back to weaker disorder under the
numerical SDRG treatment.
