9789811002755_

The purpose of this book is to introduce two recent topics in mathematical physics and probability theory: the SchrammLoewner evolution (SLE) and interacting particle systems related to random matrix theory. A typical example of the latter systems is Dysons Brownian motion (BM) model. The SLE and Dysons BM model may be considered as children of the Bessel process with parameter D, BES(D), and the SLE and Dysons 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, Cardys formula in Carlesons form is derived for SLE. In Chap. 3 Dysons BM model with parameter is introduced as a multivariate extension of BES(D) with the relation D 1. The book concentrates on the case where 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 TracyWidom distribution is derived.

...(*_*)9789811002755_

The purpose of this book is to introduce two recent topics in mathematical physics and probability theory: the SchrammLoewner evolution (SLE) and interacting particle systems related to random matrix theory. A typical example of the latter systems is Dysons Brownian motion (BM) model. The SLE and Dysons BM model may be considered as "children" of the Bessel process with parameter D, BES(D), and the SLE and Dysons 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, Cardys formula in Carlesons form is derived for SLE. In Chap. 3 Dysons BM model with parameter is introduced as a multivariate extension of BES(D) with the relation D 1. The book concentrates on the case where 2 and calls this case simply the Dyson model.

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Bessel Processes, SchrammLoewner Evolution, and the Dyson Model

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