Convergence of percolation-decorated triangulations to SLE and LQG

Convergence of percolation-decorated triangulations to SLE and LQG

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Nina Holden, MIT
Fine Hall 110

The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun. We then discuss works in progress with the goal of strengthening the topology of convergence of RPM to LQG by considering conformal embeddings of the RPM into the complex plane. This is joint with Garban, Gwynne, Miller, Sepulveda, Sheffield, and Sun.