Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric

Abstract

Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter γ$$\gamma $$, and it has long been believed that when γ=8/3$$\gamma = \sqrt{8/3}$$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other’s structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other’s structure and showing that the resulting laws agree. The present work considers a growth process called quantum Loewner evolution (QLE) on a 8/3$$\sqrt{8/3}$$-LQG surface S$${\mathcal {S}}$$ and defines dQ(x,y)$$d_{{\mathcal {Q}}}(x,y)$$ to be the amount of time it takes QLE to grow from x∈S$$x \in {\mathcal {S}}$$ to y∈S$$y \in {\mathcal {S}}$$. We show that dQ(x,y)$$d_{{\mathcal {Q}}}(x,y)$$ is a.s. determined by the triple (S,x,y)$$({\mathcal {S}},x,y)$$ (which is far from clear from the definition of QLE) and that dQ$$d_{{\mathcal {Q}}}$$ a.s. satisfies symmetry (i.e., dQ(x,y)=dQ(y,x)$$d_{{\mathcal {Q}}}(x,y) = d_{{\mathcal {Q}}}(y,x)$$) for a.a. (x, y) pairs and the triangle inequality for a.a. triples. This implies that dQ$$d_{{\mathcal {Q}}}$$ is a.s. a metric on any countable sequence sampled i.i.d. from the area measure on S$${\mathcal {S}}$$. We establish several facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric a.s. extends uniquely and continuously to the entire 8/3$$\sqrt{8/3}$$-LQG surface and that the resulting measure-endowed metric space is TBM.