Bessel Processes, Schramm-Loewner Evolution, and the Dyson Model

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 "e;children"e; of the Bessel process with parameter D, BES(D), and the SLE and Dyson's BM model as "e;grandchildren"e; 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 Tracy-Widom distribution is derived.