Existence and uniqueness of nonsimple multiple SLE

TL;DR

This paper establishes the existence and uniqueness of multiple Schramm-Loewner Evolution (SLE) processes for certain parameter ranges, advancing the mathematical understanding of these stochastic models in conformal field theory.

Contribution

It proves the existence and uniqueness of multiple SLE$_ppa$ for link patterns when ppa in (4,6], and also for ppa in (6,8), constructing the measures via an inductive approach.

Findings

Proved existence and uniqueness of multiple SLE$_ppa$ for ppa in (4,6] and (6,8).

Constructed multiple SLE measures by inductive measure construction and normalization.

Demonstrated that the total measure satisfies conformal covariance, asymptotics, and PDE properties.

Abstract

We prove the existence and uniqueness of multiple SLE$_\kappa$ associated with any given link pattern for $\kappa\in (4,6]$. We also have the uniqueness for $\kappa\in (6,8)$. The multiple SLE$_\kappa$ law is constructed by first inductively constructing a $\sigma$-finite multiple SLE$_\kappa$ measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple SLE$_\kappa$ partition functions in the literature subject to the assumption that it is smooth.