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

1. Auflage 2016
ISBN: 978-981-10-0274-8
Verlag: Springer

Buch, Englisch, Band 11, 141 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 2409 g

Reihe: SpringerBriefs in Mathematical Physics

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.

Zielgruppe

Research

Autoren/Hrsg.

Katori, Makoto

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

Preface.- 1 Bessel Process.- 2 Schramm-Loewner Evolution (SLE).- 3 Dyson Model.- References.- Solutions.- Index.