Convergence of square tilings to the Riemann map

Abstract

A well-known theorem of Rodin \& Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $Ω$ into the unit disc $\mathbb{D}$ converges to a Riemann map from $Ω$ to $\mathbb{D}$ when the mesh size converges to 0. We prove the analogous statement when circle packings are replaced by the square tilings of Brooks et al.