Geometry of random potentials: Induction of two-dimensional gravity in quantum Hall plateau transitions

Integer quantum Hall plateau transitions are usually modeled by a system of noninteracting electrons moving in a random potential. The physics of the most relevant degrees of freedom, the edge states, is captured by a recently proposed random network model, in which randomness is induced by a parameter-dependent modification of a regular network. In this paper we formulate a specific map from random potentials onto two-dimensional (2D) discrete surfaces, which indicates that 2D gravity emerges in all quantum phase transitions characterized by the presence of edge states in a disordered environment. We also establish a connection between the parameter in the network model and the Fermi level in the random potential.