Point Processes and Multiple SLE/GFF Coupling

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 $β>0$, DYS$_β$, which is regarded as multivariate extensions of BES$_D$ with the relation $β=D-1$. 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 $κ>0$, SLE$_κ$, 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$_κ$ as a complexification of BES$_D$ with the relation $κ=4/(D-1)$. The last topic of lectures is the construction of the multiple SLE$_κ$, 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$_β$ with the relation $β=8/κ$.

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