The Tutte Embedding of the Poisson\u2013Voronoi Tessellation of the Brownian Disk Converges to sqrt{8/3}-Liouville Quantum Gravity

Abstract

Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a sqrt{8/3}-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on mathbb {R}^2 is the epsilon rightarrow 0 limit of simple random walk on epsilon mathbb {Z}^2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the lambda rightarrow infty limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is lambda times the associated area measure. Among other things, this implies that as lambda rightarrow infty the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to sqrt{8/3}-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.