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Book synopsis

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 _ 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.

Book details

Book category Books in English Mathematics & science Mathematics Probability & statistics

• Full title: Bessel Processes, Schramm-Loewner Evolution, and the Dyson Model
• Author: Makoto Katori
• Language: English
• Binding: Paperback
• Number of pages: 141
• EAN: 9789811002748
• ISBN: 9811002746
• ID: 02859360
• Publisher: Springer Verlag, Singapore
• Weight: 258 g
• Dimensions: 242 × 156 × 17 mm
• Date of publishing: 29. April 2015

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