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 yields a measure-valued process $$t\mapsto \{\alpha _{N,t}\},$$ and we are interested in the limit behaviour as $$N\rightarrow \infty .$$ We prove tightness of the sequence $$\{\alpha _{N,t}\}_{N\in \mathbb {N}}$$ under certain assumptions and address some further problems. Moreover, we investigate a similar situation in which all slits are trajectories of a certain quadratic differential.