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We study the adjacency graph of bubbles---i.e., complementary connectedcomponents---of an SLE$_{\kappa}$ curve for $\kappa \in (4,8)$, with two suchbubbles considered to be adjacent if their boundaries intersect. We show thatthis 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 toa problem posed by Duplantier, Miller and Sheffield (2014). Our proof in factyields a stronger connectivity result for $\kappa \in (4,\kappa_0]$, which saysthat 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 ofindependent $\kappa/4$-stable processes, which allows us to reduce our problemto a problem about stable processes. In fact, due to this encoding, our resultscan be re-phrased as statements about the connectivity of the adjacency graphof 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 theencoding as a black box.

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