Schnyder woods, SLE(16), and Liouville quantum gravity

Abstract

In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. In the framework of mating of trees, a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple of random walks. We show that these three walks converge in the scaling limit to three Brownian motions produced in the mating-of-trees framework by Liouville quantum gravity (LQG) with parameter $1$, decorated with a triple of SLE${}_{16}$'s curves. These three SLE${}_{16}$'s curves are coupled such that the angle difference between them is $2\pi /3$ in imaginary geometry. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE.

arXiv commentFinal version; accepted at Transactions of the AMS

References in corpus (3)

• Random trees and applications
• Uniqueness and universality of the Brownian map
• Bijections for Baxter Families and Related Objects

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