The purpose of this book is to introduce two recent topics in mathematical physics and
probability theory: the Schramm-Loewner evolution (SLE) and interacting particle systems
related to random matrix theory. A typical example of the latter systems is Dyson's Brownian
motion (BM) model. The SLE and Dyson's BM model may be considered as children of the Bessel
process with parameter D BES(D) and the SLE and Dyson's BM model as grandchildren of BM. In
Chap. 1 the parenthood of BM in diffusion processes is clarified and BES(D) is defined for any
D 1. Dependence of the BES(D) path on its initial value is represented by the Bessel flow. In
Chap. 2 SLE is introduced as a complexification of BES(D). Rich mathematics and physics
involved in SLE are due to the nontrivial dependence of the Bessel flow on D. From a result for
the Bessel flow Cardy's formula in Carleson's form is derived for SLE. In Chap. 3 Dyson's BM
model with parameter beta is introduced as a multivariate extension of BES(D) with the relation
D = beta + 1. The book concentrates on the case where beta = 2 and calls this case simply the
Dyson model.The Dyson model inherits the two aspects of BES(3) hence it has very strong
solvability. That is the process is proved to be determinantal in the sense that all
spatio-temporal correlation functions are given by determinants and all of them are controlled
by a single function called the correlation kernel. From the determinantal structure of the
Dyson model the Tracy-Widom distribution is derived.
