The Minkowski Content Measure for the Liouville Quantum Gravity Metric

by Jinwoo Sung (University of Chicago & IHES)

Friday Apr 28, 2023, 3:00 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Probability and analysis informal seminar

A Liouville quantum gravity (LQG) surface is a random two-dimensional "Riemannian manifold" that is conjectured to be the scaling limit of a wide variety of random planar graph models. LQG was formulated initially as a random measure space and, more recently, as a random metric space. In this talk, I will explain how the LQG measure can be recovered as the Minkowski content measure for the LQG metric, thereby providing a direct connection between the two formulations for the first time. Our primary tool is the mating-of-trees theory of Duplantier, Miller, and Sheffield, which says that an LQG surface explored by an independent space-filling Schramm–Loewner evolution (SLE) curve is an infinitely divisible metric measure space when. This is joint work with Ewain Gwynne (University of Chicago).

 

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