Makoto Katori's Books

Elliptic Extensions in Statistical and Stochastic Systems

By: Makoto Katori

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Elliptic Extensions in Statistical and Stochastic Systems

By: Makoto Katori

Hermite's theorem makes it known that there are three levels of mathematical frames in which a simple addition formula is valid. They are rational, q-analogue, and elliptic-analogue. Based on the addition formula and associated mathematical structures, productive studies have been carried out in the process of q-extension of the rational (classical) formulas in enumerative combinatorics, theory of special functions, representation theory, study of integrable systems, and so on. Originating from the paper by Date, Jimbo, Kuniba, Miwa, and Okado on the exactly solvable statistical mechanics models using the theta function identities (1987), the formulas obtained at the q-level are now extended to the elliptic level in many research fields in mathematics and theoretical physics. In the present monograph, the recent progress of the elliptic extensions in the study of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory is shown. At the elliptic level, many special functions are used, including Jacobi's theta functions, Weierstrass elliptic functions, Jacobi's elliptic functions, and others. This monograph is not intended to be a handbook of mathematical formulas of these elliptic functions, however. Thus, use is made only of the theta function of a complex-valued argument and a real-valued nome, which is a simplified version of the four kinds of Jacobi's theta functions. Then, the seven systems of orthogonal theta functions, written using a polynomial of the argument multiplied by a single theta function, or pairs of such functions, can be defined. They were introduced by Rosengren and Schlosser (2006), in association with the seven irreducible reduced affine root systems. Using Rosengren and Schlosser's theta functions, non-colliding Brownian bridges on a one-dimensional torus and an interval are discussed, along with determinantal point processes on a two-dimensional torus. Their scaling limits are argued, and the infinite particle systems are derived. Such limit transitions will be regarded as the mathematical realizations of the thermodynamic or hydrodynamic limits that are central subjects of statistical mechanics.

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134

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Springer Nature

Published Date

ISBN 10

9811995273

ISBN 13

9789811995279

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

By: Makoto Katori

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

By: Makoto Katori

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.

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149

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Springer

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ISBN 10

9811002754

ISBN 13

9789811002755

Elliptic Extensions in Statistical and Stochastic Systems

By: Makoto Katori

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Elliptic Extensions in Statistical and Stochastic Systems

By: Makoto Katori

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

134

Publisher

Springer Nature

Published Date

ISBN 10

9811995273

ISBN 13

9789811995279

Book's by Makoto Katori

Elliptic Extensions in Statistical and Stochastic Systems

Elliptic Extensions in Statistical and Stochastic Systems

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

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

Elliptic Extensions in Statistical and Stochastic Systems

Elliptic Extensions in Statistical and Stochastic Systems

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