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

Beliaev, D
Viklund, F

1 June 2013

## Journal:

Communications in Mathematical Physics

## Last Updated:

2020-06-06T05:59:59.673+01:00

2

320

## DOI:

10.1007/s00220-013-1710-5

379-394

## 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.

190373

Not Submitted

Journal Article