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# Bessel Processes, Schramm–loewner Evolution, and the Dyson Model (Paperback) (Makoto Katori)

### About this item

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 ß is introduced as a multivariate extension of BES(*D*) with the relation*D*= ß + 1. The book concentrates on the case where ß = 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.

**Genre:**Science, Mathematics

**Sub-Genre:**Probability + Statistics / General, Mathematical Physics, Physics

**Series Title:**Springerbriefs in Mathematical Physics

**Format:**Paperback

**Publisher:**Springer Verlag

**Author:**Makoto Katori

**Language:**English

**Street Date**: February 16, 2016

**TCIN**: 51018623

**UPC**: 9789811002748

**Item Number (DPCI)**: 248-14-4486

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