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May 2020 Connectivity properties of the adjacency graph of $\mathrm{SLE}_{\kappa}$ bubbles for $\kappa\in(4,8)$
Ewain Gwynne, Joshua Pfeffer
Ann. Probab. 48(3): 1495-1519 (May 2020). DOI: 10.1214/19-AOP1402

Abstract

We study the adjacency graph of bubbles, that is, complementary connected components of a $\mathrm{SLE}_{\kappa }$ curve for $\kappa \in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect. We show that this adjacency graph is a.s. connected for $\kappa \in (4,\kappa _{0}]$, where $\kappa _{0}\approx 5.6158$ 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 $\kappa \in (4,\kappa _{0}]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$. We also show that there is a (nonexplicit) $\kappa _{1}\in (\kappa _{0},8)$ such that this stronger condition does not hold for $\kappa \in [\kappa _{1},8)$.

Our proofs are based on an encoding of $\mathrm{SLE}_{\kappa }$ in terms of a pair of independent $\kappa /4$-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 rephrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called $\kappa /4$-stable looptrees, as studied, for example, 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.

Citation

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Ewain Gwynne. Joshua Pfeffer. "Connectivity properties of the adjacency graph of $\mathrm{SLE}_{\kappa}$ bubbles for $\kappa\in(4,8)$." Ann. Probab. 48 (3) 1495 - 1519, May 2020. https://doi.org/10.1214/19-AOP1402

Information

Received: 1 April 2018; Revised: 1 July 2019; Published: May 2020
First available in Project Euclid: 17 June 2020

zbMATH: 07226368
MathSciNet: MR4112722
Digital Object Identifier: 10.1214/19-AOP1402

Subjects:
Primary: 60G52, 60J67

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 3 • May 2020
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