[Paper Review] Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Ewain Gwynne, Nina Holden+1 more|arXiv (Cornell University)|May 16, 2019

Stochastic processes and statistical mechanics31 references6 citations

TL;DR

This paper establishes the first joint convergence of random triangulations decorated by critical site percolation to $\sqrt{8/3}$-Liouville quantum gravity (LQG) decorated by SLE$_6$ in both the metric and peanosphere senses. Using a coupling of the metric and tree-based encoding, the authors prove that the full percolation interface collection converges to CLE$_6$ in the Gromov–Hausdorff–Prokhorov–uniform (GHPU) topology, while the associated random walk encoding converges to correlated Brownian motion, confirming simultaneous convergence in both scaling limits.

ABSTRACT

Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE$_6$ in two different ways: 1. The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov-Hausdorff topology. 2. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE$_6$-decorated $\sqrt{8/3}$-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called $ extit{peanosphere convergence}$. We prove that one in fact has $ extit{joint}$ convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to $\sqrt{8/3}$-LQG decorated by CLE$_6$ in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into $\mathbb C$ via the so-called $ extit{Cardy embedding}$ converge to $\sqrt{8/3}$-LQG.

Motivation & Objective

• To establish simultaneous convergence of random triangulations with critical site percolation in both the metric and peanosphere senses.
• To extend metric convergence from a single percolation interface to the full collection of interfaces, showing convergence to CLE$_6$-decorated $\sqrt{8/3}$-LQG.
• To provide a foundational step toward proving convergence of the Cardy embedding of random triangulations to conformal embeddings in the continuum.
• To demonstrate that the joint scaling limit captures both geometric (metric) and combinatorial (tree encoding) structures simultaneously.

Proposed method

• Utilizes a coupling of the metric and tree-based encoding to simultaneously control both the Gromov–Hausdorff–Prokhorov–uniform (GHPU) topology and the peanosphere convergence.
• Employs the mating-of-trees bijection to encode site-percolated triangulations via two-dimensional random walks, which converge to correlated Brownian motion.
• Applies the Gromov–Hausdorff–Prokhorov–uniform (GHPU) topology to show convergence of the metric measure space with curve decoration.
• Uses the Schaeffer bijection and its generalizations to relate graph distances to tree labels, enabling control over the metric structure.
• Applies the Duplantier–Miller–Sheffield mating-of-trees theorem to link the random walk encoding to SLE$_6$-decorated $\sqrt{8/3}$-LQG.
• Employs re-rooting invariance and loop ensemble techniques to couple multiple space-filling explorations and prove joint convergence of crossing events.

Experimental results

Research questions

• RQ1 Can random triangulations with critical site percolation converge jointly in both the metric and peanosphere senses to $\sqrt{8/3}$-LQG with SLE$_6$?
• RQ2 Does the full collection of percolation interfaces converge to CLE$_6$ in the metric space sense, not just a single interface?
• RQ3 Can multiple space-filling explorations from different root edges be jointly coupled in the scaling limit to preserve topological and probabilistic structure?
• RQ4 How does the convergence of random walk encodings relate to the convergence of the underlying geometric and curve-decorated metric structures?
• RQ5 Can the joint convergence be used to prove convergence of the Cardy embedding to the conformal embedding in the continuum?

Key findings

• The paper establishes the first joint convergence of any random planar map model in both the metric and peanosphere senses.
• The full collection of percolation interfaces converges to CLE$_6$-decorated $\sqrt{8/3}$-LQG in the GHPU topology, extending prior results that only considered a single interface.
• The random walk encoding of the percolated triangulation converges to a correlated two-dimensional Brownian motion, confirming peanosphere convergence.
• Joint convergence of multiple space-filling explorations from different root edges is proven, enabling the convergence of crossing events to their continuum analogs.
• The convergence of the Cardy embedding to the conformal embedding is established in [HS19], building on the joint scaling limit proven here.
• The coupling of the metric and encoding processes ensures that both geometric and combinatorial structures are preserved in the continuum limit.