The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to $\sqrt{8/3}$-Liouville quantum gravity

TL;DR

This paper proves that Brownian motion on Brownian surfaces can be approximated by simple random walks on fine Voronoi tessellations, and shows the convergence of the Tutte embedding to the continuum conformal structure of the Brownian disk.

Contribution

It establishes the limit of simple random walk on Voronoi tessellations as the Brownian motion on Brownian surfaces and proves the convergence of the Tutte embedding to the continuum conformal embedding.

Findings

Brownian motion on Brownian surfaces is the limit of random walks on fine Voronoi tessellations.

Tutte embeddings of discretized Brownian disks converge to the continuum conformal embedding.

Shapes of embedded metric balls and Voronoi cells are unlikely to be very elongated, with bounded moment estimates.

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 $\epsilon \to 0$ 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 $\lambda \to \infty$ 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 $\lambda \to \infty$ 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.