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${}_D$, $D\ge 1$, 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 $\kappa =4\mathrm{/}(D-1)$. 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 $\beta =8\mathrm{/}\kappa$.