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Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity
About this Title
Olivier Bernardi, Nina Holden and Xin Sun
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 289, Number 1440
ISBNs: 978-1-4704-6699-2 (print); 978-1-4704-7626-7 (online)
DOI: https://doi.org/10.1090/memo/1440
Published electronically: August 24, 2023
Keywords: Liouville quantum gravity,
Schramm-Loewner evolutions,
mating of trees,
percolation,
random planar maps,
triangulations,
random walks,
Kreweras walks,
planar Brownian motion,
bijection
Table of Contents
Chapters
- 1. Introduction
- 2. A bijection between Kreweras walks and percolated triangulations
- 3. Discrete dictionary I: Spine-looptrees decomposition
- 4. Discrete dictionary II: Exploration tree
- 5. Discrete dictionary III: tree of clusters, envelope excursions, and pivotal points
- 6. Proofs of the bijective correspondences
- 7. The mating-of-trees correspondence
- 8. Convergence of percolated triangulations to $CLE_6$ on $\sqrt {8/3}$-LQG
- 9. Proofs of the scaling limit results
- A. Recovering the DFS tree and the walk from the percolated UIPT
Abstract
We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt {8/3}$-LQG and SLE$_6$. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of $\sqrt {8/3}$-LQG and SLE$_6$. For instance, we show that the exploration tree of the percolation converges to a branching SLE$_6$, and that the collection of percolation cycles converges to the conformal loop ensemble CLE$_6$. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.- Louigi Addario-Berry and Marie Albenque, The scaling limit of random simple triangulations and random simple quadrangulations, Ann. Probab. 45 (2017), no. 5, 2767–2825. MR 3706731, DOI 10.1214/16-AOP1124
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