Abstract

In my talk I will discuss a general approach to better understand the geometry of Liouville quantum gravity (LQG). The idea, roughly speaking, is to partition the random surface into dyadic squares of roughly the same ``LQG size’’.  Based on this approach, I will introduce two different models of LQG that will provide answers to three questions in the field:

1) Rigorously explain the so-called ``DDK ansatz’’ by proving that, for a surface with metric tensor some regularized version of the LQG metric tensor $\exp(\gamma h) (dx^{2 + dy}2)$, its law corresponds to sampling a surface with probability proportional to $(\det_{\zeta}’ \Delta)^{-c/2}$, with $c$ the matter central charge.

2) Provide a heuristic picture of the geometry of LQG with matter central charge in the interval $(1,25)$. (The geometry in this regime is mysterious even from a physics perspective.)

3) Explain why many works in the physics literature may have missed the nontrivial conformal geometry of LQG with matter central charge in the interval $(1,25)$ when they suggest (based on numerical simulations and heuristics) that LQG exhibits the macroscopic behavior of a continuum random tree in this phase.

This talk is based on a joint work with Morris Ang, Minjae Park, and Scott Sheffield; and a joint work with Ewain Gwynne, Nina Holden, and Guillaume Remy.