Quantum Hall transitions on random networks and exact results from quantum gravity

The paradigmatic Chalker-Coddington network model for the integer quantum Hall transition (QHT) was generalized to random networks. Numerical studies of the random networks show that critical exponents of the integer QHT are modified by the geometric randomness. It was conjectured that the changes are similar to the ones induced by random geometry (two-dimensional quantum gravity) at critical points of conventional statistical mechanics models (Ising, Potts, O(N), etc.) and described by the so called Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling relation. Here we investigate these issues for the spin QHT which can be mapped to classical percolation. The mapping works even in the presence of geometric disorder, and we solve the spin QHT on random networks as percolation on certain random graphs using methods of discrete quantum gravity (matrix models and loop equations). We confirm that the KPZ scaling works in this case and determine various exact critical exponents for the spin QHT on random networks. We also discuss how our findings are related to the (violation of the) Harris criterion and the Chase-Chase-Fisher-Spencer inequality.