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

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.