Multiple SLE and the complex Burgers equation

TL;DR

This paper investigates the limit of multiple Schramm-Loewner Evolutions (SLE) as the number of slits increases infinitely, revealing a connection to a deterministic Loewner equation governed by a complex Burgers equation.

Contribution

It introduces a novel limit process for multiple SLEs leading to a deterministic Loewner equation linked to the complex Burgers equation.

Findings

Limit of multiple SLEs yields a deterministic Loewner equation.

The vector field in the limit is governed by a complex Burgers equation.

Provides a new perspective on the continuum limit of multiple SLEs.

Abstract

In this paper we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of $n$ slits that connect $n$ points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.