Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity

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 matingof-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 ?8/3-LQG and SLE6. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of ?8/3-LQG and SLE6. For instance, we show that the exploration tree of the percolation converges to a branching SLE6, and that the collection of percolation cycles converges to the conformal loop ensemble CLE6. 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.

Author: Olivier Bernardi
Format: Paperback / Soft Cover
Pages: 176
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