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Elliptic Extensions in Statistical and Stochastic Systems

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 47)

DOI: 10.1007/978-981-19-9527-9

ISBN: 978-981-19-9526-2

Publisher: Springer Nature Singapore - Springer Journals

Apr 7, 2023

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Chapter 1Introduction

Katori, Makoto

Pages: 1 –7

[The q-extension is the procedure to replace mathematical symbols, identities, functions and others by their meaningful q-analogues, and has been extensively studied. As a typical example of q-extension, here we introduce the q-Pochhammer symbol as the q-analogue of the well-known Pochhammer...
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Chapter 2Brownian Motion and Theta Functions

Katori, Makoto

Pages: 9 –19

[We introduce the Brownian motion on a real line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
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Chapter 3Biorthogonal Systems of Theta Functions and Macdonald Denominators

Katori, Makoto

Pages: 21 –33

[The fact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{p \rightarrow...
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Chapter 4KMLGV Determinants and Noncolliding Brownian Bridges

Katori, Makoto

Pages: 35 –57

[In Chap. 2, for the one-dimensional torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
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Chapter 5Determinantal Point Processes Associated with Biorthogonal Systems

Katori, Makoto

Pages: 59 –80

[The seven types of noncolliding Brownian bridges constructed so far can be regarded as the time-dependent point processes on the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...
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Chapter 6Doubly Periodic Determinantal Point Processes

Katori, Makoto

Pages: 81 –100

[We consider the orthogonal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
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Chapter 7Future Problems

Katori, Makoto

Pages: 101 –108

[So far we have studied two kinds of random systems whose probability laws are expressed using the theta function. One of them is a family of one-dimensional stochastic processes consisting of seven types of noncolliding Brownian bridges. Another one is a family of two-dimensional point processes...

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