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

Makoto Katori (Autor)

Buch | Softcover

141 Seiten

2016 | 1st ed. 2016
Springer Verlag, Singapore
978-981-10-0274-8 (ISBN)

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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.

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

Erscheinungsdatum	08.02.2016
Reihe/Serie	SpringerBriefs in Mathematical Physics ; 11
Zusatzinfo	16 Illustrations, color; X, 141 p. 16 illus. in color.
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
	Naturwissenschaften ► Physik / Astronomie ► Thermodynamik
Schlagworte	Complexifications of random variables • Conformal transformations and conformal invariance • Itô’s formula and Stochastic Calculus • Multivariate extensions and special functions • Random matrix theory and conformal field theory
ISBN-10	981-10-0274-6 / 9811002746
ISBN-13	978-981-10-0274-8 / 9789811002748
Zustand	Neuware