Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding

Abstract

We endow the 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere S 2 . Previous work in this series used quantum Loewner evolution (QLE) to construct a metric d Q on a countable dense subset of S 2 . Here we show that d Q a.s. extends uniquely and continuously to a metric d Q on all of S 2 . Letting d denote the Euclidean metric on S 2 , we show that the identity map between (S 2 , d) and (S 2 , d Q ) is a.s. Hölder continuous in both directions. We establish several other properties of (S 2 , d Q ), culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of (S 2 , d Q ) geodesics.

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

“…where dx denotes Lebesgue measure on ∂D. (One can also associate with γ-LQG a canonical metric [36,34,7,14], but we will not need this in the present paper.) If (D, h) and ( D, h) are related by ψ as in (2.8), then for all A ⊆ D, µ h (A) = µ h (ψ(A)), that is, µ h is the push-forward µ h by ψ.…”

The Brownian Map

“…Nonetheless, one can construct a random compact metric space based on Liouville quantum gravity-and when γ = 8/3, and Σ = S 2 , this gives the Brownian map! This fact has been shown in a number of ways, starting with Jason Miller and Scott Sheffield [8] around 2016. More recently, Holden and Xin Sun showed how to go the other way, and recover the Gaussian free field on the sphere from large uniform triangulations [2].…”

The volume measure of the Brownian sphere is a Hausdorff measure

“…This volume measure is expected to appear as the limit of (scaled) counting measures on vertices when the Brownian sphere is written as the Gromov-Hausdorff-Prokhorov limit of large random planar maps (see [20,Theorem 7] for the case of quadrangulations). We mention that a very different approach to the Brownian sphere, involving deep connections with Liouville quantum gravity has been developed by Miller and Sheffield in a series of papers [25,26,27,28]. More recently, Ding, Dubédat, Dunlap and Falconet [11] have studied Liouville first-passage percolation metrics associated with mollified versions of the Gaussian free field and were able to prove the tightness of these renormalized metrics.…”

“…Although this result seemed plausible, it was not obvious from the construction of the Brownian sphere in terms of Brownian motion indexed by the Brownian tree. We note that other models of random geometry such as the Brownian plane and the Brownian disk have been investigated in recent papers (see in particular [8,9,10,20]) and also correspond to specific quantum surfaces, in the terminology of [26,27,28] (see [27,Corollary 1.5]). It is not hard to verify that Theorem 1 can be extended to these models, using the known connections between them and the Brownian sphere.…”