Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions

TL;DR

This paper proves that solutions to a multiple Loewner equation driven by Dyson's Brownian motions generate multiple continuous curves with properties depending on the parameter , advancing understanding of multiple SLEs.

Contribution

It establishes that the multiple Loewner equation solutions produce multiple curves and characterizes their nature across different  regimes, using Gaussian free field coupling.

Findings

Solutions generate multiple continuous curves.

Curve types depend on : simple, intersecting, or space-filling.

Confirmed longstanding conjecture about multiple SLEs.

Abstract

The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $\kappa\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the solution of the multiple Loewner equation generates multiple curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and multiple SLEs. Consequently, we prove the longstanding conjecture that the solution indeed generates multiple continuous curves. Furthermore, these multiple curves are (i) simple disjoint curves when $\kappa\in (0,4]$, (ii) intersecting curves when $\kappa\in (4,8)$, and (iii) space-filling curves when $\kappa=8$.