Skip to main content
SpringerLink
Log in
Book cover

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

  • Book
  • © 2016

Overview

Authors:
  1. Makoto Katori

    1. Department of Physics, Chuo University, Tokyo, Japan

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • 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
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 11)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (3 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Bessel Processes

    • Makoto Katori
    Pages 1-39

  3. Schramm–Loewner Evolution (SLE)

    • Makoto Katori
    Pages 41-56

  4. Dyson Model

    • Makoto Katori
    Pages 57-137

  5. Back Matter

    Pages 139-141
    Download chapter PDF
Back to top

Keywords

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. 

Authors and Affiliations

  • Department of Physics, Chuo University, Tokyo, Japan

    Makoto Katori

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation