Mathematics – Probability
Scientific paper
2010-12-23
Mathematics
Probability
Rewritten and expanded with several new results; 15 pages
Scientific paper
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.
Beliaev Dmitry
Viklund Fredrik Johansson
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