Point Processes and Multiple SLE/GFF Coupling

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 $\beta=8/\kappa$.