Perspectives on discrete models of supercritical Liouville quantum gravity

Abstract: Liouville quantum gravity (LQG) is a one-parameter model of surfaces with random geometry. The parameter, called the central charge c, controls how fractal the LQG surface is. Whereas the subcritical and critical phases of LQG, corresponding to c≥25, has been investigated extensively in recent years, most of the available tools (Gaussian multiplicative chaos, mating of trees, etc) do not extend to the supercritical phase c∈(1,25). 

In this talk, I will present a random planar map model of supercritical LQG, based on the coupling with CLE(4) by Ang and Gwynne, and consider its possible scaling limits. A priori, it can have infinitely many vertices with positive probability; conditioning on the event that this model is finite, we obtain the continuum random tree as the scaling limit as predicted in some of the physics literature. Without this conditioning, we expect it to have a different scaling limit described by the Gaussian free field (GFF). To this end, we formulate a precise scaling limit conjecture for the Tutte embedding of this model, defined using a novel construction of random walks on graphs reflected off of its infinite ends. This is joint work with Manan Bhatia (MIT) and Ewain Gwynne (UChicago).