Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity
Olivier Bernardi, Nina Holden, Xin Sun
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 <italic>mating-of-trees</italic> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 8 slash 3 EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>8</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {8/3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -LQG and SLE <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 6"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 8 slash 3 EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>8</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {8/3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -LQG and SLE <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 6"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For instance, we show that the exploration tree of the percolation converges to a branching SLE <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 6"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and that the collection of percolation cycles converges to the conformal loop ensemble CLE <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 6"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.