Point Processes and Multiple SLE/GFF Coupling

TL;DR

This paper explores the probability laws of random points, curves, and surfaces, focusing on Dyson's Brownian motion, Gaussian analytic functions, and Schramm--Loewner evolution, culminating in the coupling of multiple SLE with Gaussian free fields.

Contribution

It establishes the precise coupling condition between multiple SLE$_{\kappa}$ and Gaussian free fields via the DYS$_{eta}$ process with a specific parameter relation.

Findings

Coupling of multiple SLE$_{\kappa}$ with GFF is characterized by the DYS$_{eta}$ process.

Relation between parameters: =8/ for the coupling to hold.

Connection between SLE, Dyson's model, and Gaussian analytic functions.

Abstract

In the series of lectures, we will discuss probability laws of random points, curves, and surfaces. Starting from a brief review of the notion of martingales, one-dimensional Brownian motion (BM), and the $D$-dimensional Bessel processes, BES$_{D}$, $D \geq 1$, first we study Dyson's Brownian motion model with parameter $\beta >0$, DYS$_{\beta}$, which is regarded as multivariate extensions of BES$_D$ with the relation $\beta=D-1$. Next, using the reproducing kernels of Hilbert function spaces, the Gaussian analytic functions (GAFs) are defined on a unit disk and an annulus. As zeros of the GAFs, determinantal point processes and permanental-determinantal point processes are obtained. Then, the Schramm--Loewner evolution with parameter $\kappa >0$, SLE$_{\kappa}$, is introduced, which is driven by a BM on ${\mathbb{R}}$ and generates a family of conformally invariant probability laws of random curves on the upper half complex plane ${\mathbb{H}}$. We regard SLE$_{\kappa}$ as a complexification of BES$_D$ with the relation $\kappa=4/(D-1)$. The last topic of lectures is the construction of the multiple SLE$_{\kappa}$, which is driven by the $N$-particle process on ${\mathbb{R}}$ and generates $N$ interacting random curves in ${\mathbb{H}}$. We prove that the multiple SLE/GFF coupling is established, if and only if the driving $N$-particle process on ${\mathbb{R}}$ is identified with DYS$_{\beta}$ with the relation $\beta=8/\kappa$.