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

Understand

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).