2018/09/06 by Ewain Gwynne, Gwynne, Ewain, Jason Miller +3 · 1 citation
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1809.02091
openalex publication_date 2018/09/06 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
Recent works have shown that an instance of a Brownian surface (such as the\nBrownian map or Brownian disk) a.s. has a canonical conformal structure under\nwhich it is equivalent to a \√(8/3)-Liouville quantum gravity (LQG)\nsurface. In particular, Brownian motion on a Brownian surface is well-defined.\nThe construction in these works is indirect, however, and leaves open a basic\nquestion: is Brownian motion on a Brownian surface the limit of simple random\nwalk on increasingly fine discretizations of that surface, the way Brownian\nmotion on mathbb R2 is the \ε \→ 0 limit of simple random walk on\n\ε mathbb Z2?\n We answer this question affirmatively by showing that Brownian motion on a\nBrownian surface is (up to time change) the \λ \→ \∞ limit of\nsimple random walk on the Voronoi tessellation induced by a Poisson point\nprocess whose intensity is \λ times the associated area measure. Among\nother things, this implies that as \λ \→ \∞ the Tutte embedding\n(a.k.a. harmonic embedding) of the discretized Brownian disk converges to the\ncanonical conformal embedding of the continuum Brownian disk, which in turn\ncorresponds to \√(8/3)-LQG.\n Along the way, we obtain other independently interesting facts about\nconformal embeddings of Brownian surfaces, including information about the\nEuclidean shapes of embedded metric balls and Voronoi cells. For example, we\nderive moment estimates that imply, in a certain precise sense, that these\nshapes are unlikely to be very long and thin.\n