Conformal restriction: The trichordal case

TL;DR

This paper extends the study of conformal restriction properties in two dimensions to the trichordal case, involving complex SLE processes with hypergeometric functions, and identifies key exponents characterizing these random sets.

Contribution

It introduces the trichordal restriction property, constructs associated random sets using specialized SLE processes, and analyzes their properties and exponents.

Findings

The trichordal restriction property is characterized and constructed.

Special variants of SLE$_{8/3}$ with drift are used for construction.

The exponent $ extstyle 20/27$ describes the law of the minimal symmetric set.

Abstract

The study of conformal restriction properties in two-dimensions has been initiated by Lawler, Schramm and Werner who focused on the natural and important chordal case: They characterized and constructed all random subsets of a given simply connected domain that join two marked boundary points and that satisfy the additional restriction property. The radial case (sets joining an inside point to a boundary point) has then been investigated by Wu. In the present paper, we study the third natural instance of such restriction properties, namely the "trichordal case", where one looks at random sets that join three marked boundary points. This case involves somewhat more technicalities than the other two, as the construction of this family of random sets relies on special variants of SLE$_{8/3}$ processes with a drift term in the driving function that involves hypergeometric functions. It turns out that such a random set can not be a simple curve simultaneously in the neighborhood of all three marked points, and that the exponent $\alpha = 20/27$ shows up in the description of the law of the skinniest possible symmetric random set with this trichordal restriction property.