Hydrodynamic Limit of Multiple SLE

TL;DR

This paper investigates the hydrodynamic limit of multiple SLE processes, deriving explicit solutions for the evolution of the SLE hulls using connections to the complex Burgers equation and Wigner's semicircle law, and explores their long-term behavior.

Contribution

It provides the first detailed analysis of the hydrodynamic limit of multiple SLE, linking it to the complex Burgers equation and Wigner's semicircle law, with explicit solutions for different initial slit configurations.

Findings

Hydrodynamic limit of multiple SLE governed by a complex Burgers equation.

Explicit solutions for SLE hull evolution starting from the origin.

Universal long-term behavior of the SLE hulls in the hydrodynamic limit.

Abstract

Recently del Monaco and Schlei{\ss}inger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on ${\mathbb{R}}$ studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, $K_t \subset {\mathbb{H}} \cup {\mathbb{R}}$ , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from $a>0$ and another half of slits start from $-a < 0$, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull $K_t$ in the hydrodynamic limit.