## Bessel Processes, Schramm–Loewner Evolution, and the Dyson ModelThe 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. |

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Bessel Processes, Schramm–Loewner Evolution, and the Dyson Model Makoto Katori No preview available - 2016 |

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absorbing Brownian motion BES(D Bessel flow Bessel process bounded function Brownian motion Brownian paths Bx(t called Chap complex Brownian motion conformal invariance conformal transformation consider correlation functions correlation kernel defined denoted Derive determinantal martingale determinantal point process determinantal process differential equation diffusion process dimension Dyson model entire function Exercise Fredholm determinant function F Gaussian given gives Hermite independent BMs inf{t infinite number initial configuration integral Itô's formula KAiry Katori Lemma local martingale Loewner chain Markov property Math matrix matrix-valued multivariate noncolliding Brownian motion number of particles O’Connell obtained one-dimensional origin particle systems Phys polynomials probability law Proof Proposition quadratic variation random matrices random matrix theory Rx(t satisfies scaling limit Sect SLE path ſº Tanemura Theorem transition probability density Widom