Conformal weldings of random surfaces: SLE and the quantum gravity zipper

Abstract

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLEdecorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.

Imaginary geometry I: interacting SLEs

“…The boundary data for h is chosen so that the central ("north-going") curve shown should approximate an SLE 2 process (color figure online) the time-reversals of SLE κ (ρ) for all values of κ, a complete construction of trees of flow lines started from interior points, the first proof that conformal loop ensembles CLE κ are canonically defined when κ ∈ (4,8), and a geometric interpretation of these loop ensembles. In subsequent works, we expect these results to be useful to the theory of Liouville quantum gravity, allowing one to generalize the results about "conformal weldings" of random surfaces that appear [31], and to complete the program outlined in [32] for showing that discrete loop-decorated random surfaces have CLE-decorated Liouville quantum gravity as a scaling limit, at least in a certain topology. We will find that many basic SLE and CLE properties can be established more easily and in more generality using the theory developed here.…”

“…2,3,4,5). Several works in recent years have addressed special cases and variants of this question [6,8,10,19,28,31,37] and have shown that in certain circumstances there is a sense in which the paths are well-defined (and uniquely determined) by h, and are variants of the Schramm-Loewner evolution (SLE). In this article, we will focus on the case that z is point on the boundary of the domain where h is defined and establish a more general set of results.…”

“…The GFF can be used to generate various kinds of random geometric structures, including both Liouville quantum gravity and the imaginary geometry discussed here [31]. Roughly speaking, the former corresponds to replacing a Euclidean metric dx 2 + dy 2 with e γ h (dx 2 + dy 2 ) [where γ ∈ (0, 2) is a fixed constant and h is the Gaussian free field].…”

“…A brief overview of imaginary geometry (as defined for general functions h) appears in [31], where the rays are interpreted as geodesics of a variant of the Levi-Civita connection associated with Liouville quantum gravity. One can interpret the e ih direction 6 The set of flow lines in D will be the pullback via a conformal map ψ of the set of flow lines in D provided h is transformed to a new function h in the manner shown as "north" and the e i(h+π/2) direction as "west", etc.…”

“…Essentially, the theorem states that if we sample a particular random curve on a domain D-and then sample a Gaussian free field on D minus that curve with certain boundary conditions-then the resulting field (interpreted as a distribution on all of D) has the law of a Gaussian free field on D with certain boundary conditions. It is proved in [6] that Theorem 1.1 holds for any κ and ρ for which a solution to (1.10) exists (this can also be extended to a continuum of force points; this is done for a time-reversed version of SLE in [31]). The special case of ±λ boundary conditions also appears in [28].…”