# Limits of radial multiple SLE and a Burgers-Loewner differential equation

## TL;DR

This paper studies the behavior of multiple radial SLE curves as their number grows large, revealing a connection to Burgers-type Loewner equations and exploring their implications in probability and complex analysis.

## Contribution

It establishes conditions for the tightness of processes in the infinite-curve limit and links the limiting behavior to Burgers differential equations within the Loewner framework.

## Key findings

- Infinite-slit limit described by a Burgers-type Loewner equation
- Conditions for tightness of multiple radial SLE processes
- Connection between Burgers equation and free probability measures

## Abstract

We consider multiple radial SLE as the number of curves tends to infinity. We give conditions that imply the tightness of the associated processes given by the Loewner equation. In the case of equal weights, the infinite-slit limit is described by a Loewner equation whose Herglotz vector field is given by a Burgers differential equation. Furthermore, we investigate a more general form of the Burgers equation. On the one hand, it appears in connection with semigroups of probability measures on the unit circle with respect to free convolution. On the other hand, the Burgers equation itself is also a Loewner differential equation for certain subordination chains.