Some remarks on SLE bubbles and Schramm's two-point observable

TL;DR

This paper rigorously proves a formula for the probability related to SLE(8/3) paths, derives explicit connectivity functions for SLE bubbles, constructs a restriction measure, and investigates the area distribution of these bubbles.

Contribution

It provides a rigorous proof of Simmons and Cardy's formula, derives explicit connectivity functions, constructs a restriction measure, and explores the area distribution of SLE(8/3) bubbles.

Findings

Proved the probability formula for SLE(8/3) paths passing to the left of two points.

Derived explicit expressions for connectivity functions of SLE(8/3) bubbles.

Numerically evaluated the area of SLE(8/3) bubbles and related it to the Airy distribution.

Abstract

Simmons and Cardy recently predicted a formula for the probability that the chordal SLE(8/3) 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 SLE(8/3) 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 SLE(8/3) 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 SLE(8/3) 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.