We analyze experimental quantum Hall data from a wide range of different materials, including semiconducting heterojunctions, thin films, surface layers, graphene, mercury telluride, bismuth antimonide, and black phosphorus. The fact that these materials have little in common, except that charge transport is effectively two-dimensional, shows how robust and universal the quantum Hall phenomenon is. The scaling and fixed point data we analyzed appear to show that magnetotransport in two dimensions is governed by a small number of universality classes that are classified by modular symmetries, which are infinite discrete symmetries not previously seen in nature. The Hall plateaux are (infrared) stable fixed points of the scaling-flow, and quantum critical points (where the wave function is delocalized) are unstable fixed points of scaling. Modular symmetries are so rigid that they in some cases fix the global geometry of the scaling flow, and therefore predict the exact location of quantum critical points, as well as the shape of flow lines anywhere in the phase diagram. We show that most available experimental quantum Hall scaling data are in good agreement with these predictions.

Frictional finger trees are patterns emerging from non-equilibrium processes in particle-fluid systems. Their formation share several properties with growth algorithms for minimum spanning trees (MSTs) in random energy landscapes. We propose that the frictional finger trees are indeed in the same geometric universality class as the MSTs, which is checked using updated numerical simulation algorithms for frictional fingers. We also propose a theoretical model for anomalous diffusion in these patterns, and discuss the role of diffusion as a tool to classify geometry.

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