  We study the low-energy physics of the critical (2+1)-dimensional random
transverse-field Ising model. The one-dimensional version of the model is a
paradigmatic example of a system governed by an infinite-randomness fixed
point, for which many results on the distributions of observables are known via
an asymptotically exact renormalization group (RG) approach. In two dimensions,
the same RG rules have been implemented numerically, and demonstrate a flow to
infinite randomness. However, analytical understanding of the structure of this
RG has remained elusive due to the development of geometrical structure in the
graph of interacting spins. To understand the character of the fixed point, we
consider the RG flow acting on a joint ensemble of graphs and couplings. We
propose that the RG effectively occurs in two stages: (1) randomization of the
interaction graph until it belongs to a certain ensemble of random
triangulations of the plane, and (2) a flow of the distributions of couplings
to infinite randomness while the graph ensemble remains invariant. This picture
is substantiated by a numerical RG in which one obtains a steady-state graph
degree distribution and subsequently infinite-randomness scaling distributions
of the couplings. Both of these aspects of the RG flow can be approximately
reproduced in simplified analytical models.
