Tightness results for infinite-slit limits of the chordal Loewner equation

TL;DR

This paper investigates the limiting behavior of multi-slit Loewner equations describing multiple SLE curves as the number of slits tends to infinity, establishing tightness of the associated measures under specific conditions.

Contribution

It proves the tightness of the measure-valued processes for the multi-slit Loewner equation as the number of slits grows large, advancing understanding of infinite-slit limits.

Findings

Established tightness of the measure-valued processes as N approaches infinity

Provided conditions under which the tightness holds

Addressed additional related problems in the context of infinite-slit limits

Abstract

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting $N$ points on $\mathbb{R}$ to infinity within the upper half-plane. For every $N\in\mathbb{N}$, this equation provides a measure valued process $t\mapsto \{\alpha_{N,t}\},$ and we are interested in the limit behaviour as $N\to\infty.$ We prove tightness of the sequence $\{\alpha_{N,t}\}_{N\in\mathbb{N}}$ under certain assumptions and address some further problems.