2022/07/28 by Katori, Makoto
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2207.14362
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, BESD, D ≥ 1, first we study Dyson's Brownian motion model with parameter β>0, DYSβ, which is regarded as multivariate extensions of BESD 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, SLEκ, is introduced, which is driven by a BM on ℝ and generates a family of conformally invariant probability laws of random curves on the upper half complex plane ℍ. We regard SLEκ as a complexification of BESD with the relation κ=4/(D-1). The last topic of lectures is the construction of the multiple SLEκ, which is driven by the N-particle process on ℝ and generates N interacting random curves in ℍ. We prove that the multiple SLE/GFF coupling is established, if and only if the driving N-particle process on ℝ is identified with DYSβ with the relation β=8/κ.