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.
Highlights
Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on R2 is the → 0 limit of simple random walk on Z2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is the λ → ∞ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is λ times the associated area measure
The limiting objects that one obtains are collectively known as Brownian surfaces
Our main result (Theorem 1.1) says that as λ → ∞, the random walk on the adjacency graph of Voronoi cells converges modulo parameterization to a limiting continuous path. This path is a Brownian motion when it is embedded into C using the identification of Brownian surfaces with 8/3-Liouville quantum gravity (LQG)
Summary
Our main result (Theorem 1.1) says that as λ → ∞, the random walk on the adjacency graph of Voronoi cells converges modulo parameterization to a limiting continuous path This path is a Brownian motion (modulo parameteriz√ation) when it is embedded into C using the identification of Brownian surfaces with 8/3-LQG. Once certain properties of our Voronoi cells have been established, this theorem shows that random walk on the Poisson–Voronoi tessellation of the 0-quantum cone conve√rges to Brownian motion modulo time parameterization. All of the arguments in this paper carry over verbatim to the γ -LQG metric for general γ ∈ (0, 2), as defined in [GM19b], except for the proof of the ball volume concentration bound in√Proposition 4.8 (which uses estimates for the Brownian map, so only works for γ = 8/3). A” contains the proofs of several elementary properties of Voronoi cells which follow from basic properties of Brownian surfaces and the GFF
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