SpringerBriefs in Mathematical Physics

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

Authors: Katori, Makoto

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  • 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 Schramm–Loewner 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
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eBook 46,00 €
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  • ISBN 978-981-10-0275-5
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Softcover 57,19 €
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  • ISBN 978-981-10-0274-8
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About this book

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

Table of contents (3 chapters)

Table of contents (3 chapters)

Buy this book

eBook 46,00 €
price for Spain (gross)
  • ISBN 978-981-10-0275-5
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 57,19 €
price for Spain (gross)
  • ISBN 978-981-10-0274-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Bessel Processes, Schramm–Loewner Evolution, and the Dyson Model
Authors
Series Title
SpringerBriefs in Mathematical Physics
Series Volume
11
Copyright
2016
Publisher
Springer Singapore
Copyright Holder
The Author(s)
eBook ISBN
978-981-10-0275-5
DOI
10.1007/978-981-10-0275-5
Softcover ISBN
978-981-10-0274-8
Series ISSN
2197-1757
Edition Number
1
Number of Pages
X, 141
Number of Illustrations
16 illustrations in colour
Topics