The purpose of this book is to introduce two recent topics in mathematical physics and probability theory: the SchrammLoewner 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 is introduced as a multivariate extension of BES( D ) with the relation D = + 1. The book concentrates on the case where = 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 TracyWidom distribution is derived.

Discusses the new trend in which complex analysis is applied, in contrast to original probability theory and statistical mechanics Includes new topics such as the SchrammLoewner evolution and the random matrix theory Relates topics not only to probability theory and statistical mechanics but also to quantum integrable systems, representation theory, enumerative combinatorics, conformal field theory, and functional analysis Includes supplementary material: sn.pub/extras

Inhalt
Preface.- 1 Bessel Process.- 2 Schramm-Loewner Evolution (SLE).- 3 Dyson Model.- References.- Solutions.- Index.