Connectivity properties of the adjacency graph of SLE$_\kappa$ bubbles for $\kappa \in (4,8)$


Type
Article
Change log
Authors
Gwynne, Ewain 
Pfeffer, Joshua 
Abstract

We study the adjacency graph of bubbles---i.e., complementary connected components---of an SLEκ curve for κ∈(4,8), with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s.\ connected for κ∈(4,κ0], where κ0≈5.6158 is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for κ∈(4,κ0], which says that there is a Markovian way of finding a path from any fixed bubble to . We also show that there is a (non-explicit) κ1∈(κ0,8) such that this stronger condition does not hold for κ∈[κ1,8).

Our proofs are based on an encoding of SLEκ in terms of a pair of independent κ/4-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be re-phrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called κ/4-stable looptrees, as studied, e.g., by Curien and Kortchemski (2014).

The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box.

Description
Keywords
4901 Applied Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences
Journal Title
Annals of Probability
Conference Name
Journal ISSN
0091-1798
Volume Title
Publisher
Institute of Mathematical Statistics
Rights
All rights reserved
Sponsorship
NSF grant DMS 1209044; National Science Foundation Graduate Research Fellowship under Grant No. 1122374