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

TL;DR

This paper investigates the connectivity of the adjacency graph of SLE$_$ bubbles for  <  < 8, establishing conditions under which the graph is almost surely connected and providing insights into the structure of these random fractal curves.

Contribution

It introduces a new connectivity result for the adjacency graph of SLE$_$ bubbles for  <   , using an encoding via stable processes, partially answering a question by Duplantier, Miller, and Sheffield.

Findings

The adjacency graph is almost surely connected for  <   .

There exists a _1  in (_0,8) where the stronger connectivity fails.

The proof uses an encoding of SLE in terms of /4-stable processes, linking to stable looptrees.

Abstract

We study the adjacency graph of bubbles---i.e., complementary connected components---of an 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 (non-explicit) $\kappa_1 \in (\kappa_0, 8)$ such that this stronger condition does not hold for $\kappa \in [\kappa_1,8)$. Our proofs are based on an encoding of 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 re-phrased 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, 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.