Random geometry at an infinite-randomness fixed point

The critical one-dimensional random transverse-field Ising model is a paradigmatic example of a system whose low-energy physics is governed by an infinite-randomness fixed point, for which many results on the distributions of couplings are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules can be implemented numerically, and demonstrate a flow to infinite randomness. However, theoretical understanding remains elusive due to the development of geometrical structure in the graph of interacting spins. To better understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings, and characterize the statistics of the geometry that emerges at the infinite-randomness fixed point, with a combination of numerical and simplified analytical RGs.