Some Remarks on SLE Bubbles and Schramm's Two-point Observable

Simmons and Cardy recently predicted a formula for the probability that the chordal SLE8/3 path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE8/3 bubbles conditioned to be of macroscopic size. By passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting SLE8/3 paths. The one-point version of this result has been predicted by Gamsa and Cardy. Finally, we derive an integral formula for the second moment of the area of an SLE8/3 bubble conditioned to have radius 1. We evaluate the area integral numerically and relate its value to a hypothesis that the area follows the Airy distribution.