# Understanding Liouville quantum gravity through two square subdivision models

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 $exp(gamma h) (dx2 + dy2)$, 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.

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