论文标题
点过程和多个SLE/GFF耦合
Point Processes and Multiple SLE/GFF Coupling
论文作者
论文摘要
在一系列讲座中，我们将讨论随机点，曲线和表面的概率定律。首先是对Martingales的概念，一维的Brownian Motion（BM）和$ D $维的Bessel过程，BES $ _ {d} $，$ d \ geq 1 $，首先，我们研究dyson的Brownian's Brownian Motion at paramet $β> $β> 0 $_β$的$ quortiv y Muthiv an $β= D-1 $。接下来，使用希尔伯特功能空间的再现核，在单位磁盘和环上定义了高斯分析函数（GAFS）。作为GAF的零，确定点过程和永久性确定点过程。然后，引入了带有参数$κ> 0 $的schramm - loewner进化，该$_κ$被引入，由$ {\ mathbb {r}} $的BM驱动，并生成了上半部复杂平面$ {\ mathbbbb {我们将SLE $_κ$视为与关系$κ= 4/（D-1）$的BES $ _D $的复杂化。讲座的最后一个主题是构建多个SLE $_κ$，该$由$ {\ Mathbb {r}} $驱动，并在$ {\ mathbb {h}} $中生成$ n $交互随机曲线。我们证明，当且仅当$ {\ mathbb {r}} $上的$ n $零件进程时，就确定了多个SLE/GFF耦合。
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 $β=8/κ$.
