Elliptic Extensions in Statistical and Stochastic Systems 9789811995262, 9789811995279

Table of contents :
Preface
Contents
Notation
1 Introduction
1.1 q-Extensions
1.2 Theta Functions and Elliptic Extensions
Exercises
2 Brownian Motion and Theta Functions
2.1 Brownian Motion on mathbbR
2.2 Brownian Motion on the One-Dimensional Torus mathbbT
2.3 Brownian Motion in the Interval [0, π]
2.3.1 Absorbing at Both Boundary Points
2.3.2 Reflecting at Both Boundary Points
2.3.3 Absorbing at One Boundary Point and Reflecting at Another Boundary Point
2.4 Expressions by Jacobi's Theta Functions
Exercise
3 Biorthogonal Systems of Theta Functions and Macdonald Denominators
3.1 An-1 Theta Functions and Determinantal Identity
3.2 Other Rn Theta Functions and Determinantal Identities
3.3 Biorthogonality of An-1 Theta Functions
3.4 Biorthogonality of Other Rn Theta Functions
Exercises
4 KMLGV Determinants and Noncolliding Brownian Bridges
4.1 Karlin–McGregor–Lindström–Gessel–Viennot (KMLGV) Determinants
4.2 Noncolliding Brownian Bridges on mathbbT
4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π]
4.4 Noncolliding Brownian Bridges and Macdonald Denominators
Exercise
5 Determinantal Point Processes Associated with Biorthogonal Systems
5.1 Correlation Functions of a Point Process
5.2 Determinantal Point Processes (DPPs)
5.3 Reductions to Trigonometric DPPs
5.4 Infinite Particle Systems
5.4.1 Diffusive Scaling Limits
5.4.2 Temporally Homogeneous Limits
5.4.3 DPPs at Time t=T/2
Exercises
6 Doubly Periodic Determinantal Point Processes
6.1 Orthonormal Theta Functions in the Fundamental Domain in mathbbC
6.2 DPPs on the Two-Dimensional Torus mathbbT2
6.3 Three Types of Ginibre DPPs
Exercises
7 Future Problems
7.1 Stochastic Differential Equations for Dynamically Determinantal Processes
7.2 One-Component Plasma Models and Gaussian Free Fields
Exercise
Appendix Solutions to Exercises
Appendix References
Index

Citation preview

SpringerBriefs in Mathematical Physics Volume 47

Series Editors Nathanaël Berestycki, Fakultät für Mathematik, University of Vienna, Vienna, Austria Mihalis Dafermos, Mathematics Department, Princeton University, Princeton, NJ, USA Atsuo Kuniba, Institute of Physics, The University of Tokyo, Tokyo, Japan Matilde Marcolli, Department of Mathematics, University of Toronto, Toronto, Canada Bruno Nachtergaele, Department of Mathematics, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan

SpringerBriefs are characterized in general by their size (50–125 pages) and fast production time (2–3 months compared to 6 months for a monograph). Briefs are available in print but are intended as a primarily electronic publication to be included in Springer’s e-book package. Typical works might include: • An extended survey of a field • A link between new research papers published in journal articles • A presentation of core concepts that doctoral students must understand in order to make independent contributions • Lecture notes making a specialist topic accessible for non-specialist readers. SpringerBriefs in Mathematical Physics showcase, in a compact format, topics of current relevance in the field of mathematical physics. Published titles will encompass all areas of theoretical and mathematical physics. This series is intended for mathematicians, physicists, and other scientists, as well as doctoral students in related areas. Editorial Board • • • • • •

Nathanaël Berestycki (University of Cambridge, UK) Mihalis Dafermos (University of Cambridge, UK / Princeton University, US) Atsuo Kuniba (University of Tokyo, Japan) Matilde Marcolli (CALTECH, US) Bruno Nachtergaele (UC Davis, US) Hal Tasaki (Gakushuin University, Japan)

• • • •

50 – 125 published pages, including all tables, figures, and references Softcover binding Copyright to remain in author’s name Versions in print, eBook, and MyCopy

Makoto Katori

Elliptic Extensions in Statistical and Stochastic Systems

Makoto Katori Department of Physics Chuo University Tokyo, Japan

ISSN 2197-1757 ISSN 2197-1765 (electronic) SpringerBriefs in Mathematical Physics ISBN 978-981-19-9526-2 ISBN 978-981-19-9527-9 (eBook) https://doi.org/10.1007/978-981-19-9527-9 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To Hiroko, Machiko, and Rieko

Preface

It is known from Hermite’s theorem that there are three levels of mathematical frames in which a simple addition formula (the Riemann–Weierstrass addition formula) is valid [78, Chapter XX, Miscellaneous Examples 38] [30]. They are rational, q-analogue, and elliptic-analogue.1 Based on the addition formula and associated mathematical structures, fruitful developments of study have been achieved 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 of Date, Jimbo, Kuniba, Miwa, and Okado on the exactly solvable statistical-mechanics models using the theta function identities [8], 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, I will show the recent progress of the elliptic extensions in the study of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory. At the elliptic level, many special functions are used, which include Jacobi’s theta functions, Weierstrass’ elliptic functions, Jacobi’s elliptic functions, and others. However, I do not want to make this monograph a handbook of mathematical formulas of these elliptic functions. Hence I have decided to use only the theta function θ (ζ ; p) of an argument ζ ∈ C and a real nome p ∈ (0, 1), which is a simplified version of the four kinds of Jacobi’s theta functions. Then we regard θ as a single variable and introduce ‘polynomials of θ ’ with the degree, say n ∈ N. Instead of considering usual polynomials consisting of {θ j }nj=0 , we consider the function space spanned by n independent functions ψ jRn , j = 1, . . . n, which can be written using a polynomial of the argument ζ multiplied by a single θ or pairs of such functions (without θ j with any power j ≥ 2). The functions {ψ jRn }nj=1 , n ∈ N were introduced by Rosengren and Schlosser [64], in association with the 1

Throughout the present monograph, ‘elliptic’ is used to indicate the complex functions which are meromorphic and doubly periodic. Notice that this word has different meanings in mathematics, for instance, in the field of partial differential equations. There ‘elliptic’ represents a property related to the two eigenvalues of a certain diagonalizable real matrix: If the two eigenvalues have the same sign, the system is said to be elliptic. In other cases, it could be hyperbolic or parabolic. vii

viii

Preface

seven irreducible reduced affine root systems denoted by Rn = An−1 , Bn , Bn∨ , Cn Cn∨ , BCn and Dn . Using the Rn theta functions {ψ jRn }nj=1 of Rosengren and Schlosser and their extensions denoted by { jRn }nj=1 , n ∈ N in this monograph, we discuss two groups of interacting particle systems, in which n indicates the total number of particles in each system and R specifies one of the seven types for each model. The first group of systems describes the noncolliding Brownian bridges on a one-dimensional torus T (i.e., a circle) or an interval, and the second one the determinantal point processes (DPPs) on a two-dimensional torus T2 . The former systems are (1 + 1)-dimensional stochastic processes and the latter ones are statistical models of stationary point configurations in two dimensions, both of which provide mathematical models for systems of physical particles called fermions, which are interacting with repulsive forces. The boundary conditions and the initial/final configurations of the noncolliding Brownian bridges, and the periodicity conditions of the DPPs are systematically changed depending on the choice of R from the seven types. Moreover, we can argue the scaling limits associated with n → ∞ and precisely define the infinite particle systems. Such limit transitions will be regarded as the mathematical realizations of thermodynamic limits or hydrodynamic limits which are central subjects of statistical mechanics. I would like to emphasize the fact that the construction and analysis of these systems can be performed using the orthogonality properties of {ψ jRn }nj=1 and { jRn }nj=1 , n ∈ N with respect to the suitable inner products [38, 39]. I believe that, thanks to the Rn theta functions of Rosengren and Schlosser, here I can show interesting aspects of elliptic extensions in statistical and stochastic systems using only one kind of special function θ . I would like to thank Christian Krattenthaler very much for his hospitality at Fakultät für Mathematik, Universität Wien, where the present study was started on the sabbatical leave from Chuo University. I express my gratitude to Michael Schlosser, Tomoyuki Shirai, Peter J. Forrester, Piotr Graczyk, Jacek Małecki, Takuya Murayama, and Taiki Endo for their valuable discussion. I also thank Syota Esaki and Shinji Koshida for their careful reading of the draft and for providing very useful comments. All suggestions given by two anonymous reviewers of the manuscript are very important and useful for improving the text and I appreciate their efforts very much. I am grateful to Gernot Akemann, Yacin Ameur, Sungsoo Byum, Nizar Demni, Nam-Gyu Kang, Taro Kimura, Hirofumi Osada, Hjalmer Rosengren, Hideki Tanemura for giving me encouragement to prepare the manuscript. I thank Masayuki Nakamura at the Editorial Department of Springer Japan for his truly kind assistance during the preparation of this monograph. The study reported in this monograph was supported by the Grant-in-Aid for Scientific Research (C) (No.26400405), (No.19K03674), (B) (No.18H01124), (A) (No.21H04432), and (S) (No.16H06338) of the Japan Society for the Promotion of Science (JSPS). This work was supported also by the JSPS Grant-in-Aid for Transformative Research Areas (A) JP22H05105. Tokyo, Japan February 2023

Makoto Katori

Contents

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 q-Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theta Functions and Elliptic Extensions . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6

2 Brownian Motion and Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Brownian Motion on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Brownian Motion on the One-Dimensional Torus T . . . . . . . . . . . . . 2.3 Brownian Motion in the Interval [0, π ] . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Absorbing at Both Boundary Points . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reflecting at Both Boundary Points . . . . . . . . . . . . . . . . . . . . . 2.3.3 Absorbing at One Boundary Point and Reflecting at Another Boundary Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Expressions by Jacobi’s Theta Functions . . . . . . . . . . . . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 14 15 16

3 Biorthogonal Systems of Theta Functions and Macdonald Denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 An−1 Theta Functions and Determinantal Identity . . . . . . . . . . . . . . . 3.2 Other Rn Theta Functions and Determinantal Identities . . . . . . . . . . 3.3 Biorthogonality of An−1 Theta Functions . . . . . . . . . . . . . . . . . . . . . . 3.4 Biorthogonality of Other Rn Theta Functions . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 KMLGV Determinants and Noncolliding Brownian Bridges . . . . . . . . 4.1 Karlin–McGregor–Lindström–Gessel–Viennot (KMLGV) Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Noncolliding Brownian Bridges on T . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 18 19 21 21 23 27 28 32 35 35 40 46

ix

x

Contents

4.4 Noncolliding Brownian Bridges and Macdonald Denominators . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 57

5 Determinantal Point Processes Associated with Biorthogonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Correlation Functions of a Point Process . . . . . . . . . . . . . . . . . . . . . . . 5.2 Determinantal Point Processes (DPPs) . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Reductions to Trigonometric DPPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Infinite Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Diffusive Scaling Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Temporally Homogeneous Limits . . . . . . . . . . . . . . . . . . . . . . 5.4.3 DPPs at Time t = T /2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 62 67 69 69 74 74 79

6 Doubly Periodic Determinantal Point Processes . . . . . . . . . . . . . . . . . . . 6.1 Orthonormal Theta Functions in the Fundamental Domain in C . . . 6.2 DPPs on the Two-Dimensional Torus T2 . . . . . . . . . . . . . . . . . . . . . . . 6.3 Three Types of Ginibre DPPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 88 95 99

7 Future Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Stochastic Differential Equations for Dynamically Determinantal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 One-Component Plasma Models and Gaussian Free Fields . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 105 108

Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Notation

1(ω) ·, · ·, ·T ·, ·[0,π] ·, · D(2π,2π|τ |) (, F , P) ( , p,λ) ( , K ,λ) ( TRn , KTRn , λ Rn (d x))

( Rn , K Rn , λ Rn (d x)) ( TR , KTR , λ R (d x)) ( TR2n , K TR2n , dz) R R ( Ginibre , KGinibre , λN )

Indicator of ω; 1(ω) = 1 if ω is satisfied, 1(ω) = 0 otherwise  ξ, φ := S φ(x)ξ(d x) = nj=1 φ(x j ) for ξ ∈ Conf(S), φ ∈ Bc (S) A An−1 Inner product for (E p,rn−1 , E p,ˆ ˆr )

Inner product for (E Rp n , E Rpˆ n ), Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , Dn Inner product defined by the integral on the fundamental domain D(2π, 2π |τ |) in C Probability space of the one-dimensional Brownian motion (Bm) Point process specified by the probability density p with respect to the reference measure λ Determinantal point process (DPP) specified by the correlation kernel K with respect to the reference measure λ Time-dependent DPP of type Rn associated with the noncolliding Brownian bridges (Bbs) with time duration T One-dimensional stationary DPP, Rn = An−1 , Bn , Cn , Dn Time-dependent DPP with an infinite number of particles, R = A, B, C, D DPP on T2 of type Rn Ginibre DPP on C of type R, R = A, C, D

xi

xii

Notation α→∞

( α , K α , λα (d x)) ⇒ ( , K , λ(d x)) a.s. B(t) B(R) Bc (S) C C× Cc (S) c c ◦ ( , K , λ) Conf(S) χ (·) D2π,2π|τ | δi j δ(·) δx dx dx dz det Det E A E p,rn−1 E Rp n √ i := −1 | Im z K (·, ·)

Weak convergence of DPP in the vague topology Almost surely One-dimensional standard Brownian motion (Bm) at time t ≥ 0 Smallest σ -field containing all intervals on R (Borel σ -field on R) Set of all bounded measurable complex functions with compact support on S Collection of all complex numbers = complex plane, C R2 Complex plane punctured at the origin, C\{0} Set of all continuous real-valued functions with compact support on  S  ab···ω Cyclic permutation c = b c···a Dilatation of DPP ( , K , λ) by factor c Configuration space of point processes Test function Fundamental domain in C; D2π,2π|τ | := {z = x + i y ∈ C : 0 ≤ x < 2π, 0 ≤ y < 2π |τ |} Kronecker’s delta Dirac’s delta function Delta (Dirac) measure with a point mass at x Lebesgue measure on R n Lebesgue n measure on R ; dx := j=1 d x j Lebesgue measure on C; dz := d Re z d Im z Determinant Fredholm determinant Expectation for the probability measure P Space of all An−1 theta functions with nome p and norm r Space of all Rn theta functions with nome p, Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , Dn Imaginary unit n × n unit matrix Imaginary part of z ∈ C Correlation kernel of DPP

Notation

Ksin KchGUE(±1/2) (σ )

(law)

= λ λT λ[0,1] λ(dx) λN (dz) M Rn (ζ ; p) N N0 ( p; p)∞ p pt , p˜ t p˜ p(s, x; t, y) pT paa par

pra

prr p˜ T pTRn

xiii

Sine (sinc) kernel for the Gaussian unitary ensemble (GUE) Correlation kernel for the chiral Gaussian unitary ensemble (chGUE) with ν = ±1/2 Number of cyclic permutations in σ Equivalence in probability law Reference measure Uniform measure on T; λT (d x) = d x/2π Uniform measure in an interval [0, π ]; λ[0,π] (d x) = d x/π n-direct product measure;  λ(dx) := nj=1 λ(d x j ) Complex standard normal distribution on 2 C; λN (dz) := e−|z| dz/π Macdonald denominator of type Rn , ζ ∈ Cn Set of all positive integers, {1, 2, . . . } Set of all nonnegative integers, {0, 1, 2, . . . } Special case symbol  of p-Pochhammer j given by ∞ j=1 (1 − p ) Probability density for the probability measure P 2 pt := e−t , p˜ t := e−4π /t p˜ := e−2πi/τ for p = e2πiτ Transition probability density (tpd) of Bm for (s, x)  (t, y) tpd of Bm on T tpd of Bm in [0, π ] with the absorbing boundary conditions both at 0 and π tpd of Bm in [0, π ] with the absorbing (resp. reflecting) boundary condition at 0 (resp. π ) tpd of Bm in [0, π ] with the reflecting (resp. absorbing) boundary condition at 0 (resp. π ) tpd of Bm in [0, π ] with the reflecting boundary conditions both at 0 and π p˜ T (s, x; t, y) :=  ∞ w w=−∞ (−1) p(s, x; t, y + 2π w) Probability density of the noncolliding Bbs of type Rn with time duration T

xiv

{ψ jRn }nj=1 { jRn }nj=1 [φ; κ] R R+ rt r = (−1)n p n/2

Notation

Rn theta functions of Rosengren and Schlosser Orthonormal Rn theta functions having doubly-quasi-periodicity on C Characteristic function of a point process Collection of all real numbers Collection of all nonnegative real numbers; R+ := [0, ∞) If n is odd, rt = rt (n) := − pn 2 t/2 ; if n is even, rt = rt (n) := pn(n+1)t/2 A Norm for { j n−1 }nj=1 and A

Re z R( , K ,λ) ρ1 ρm S Sn Su ( , K ,λ) sgn(σ ) SO(n) Sp(n) T T2 θ (z; p) ϑ j (ξ ; τ )

U(n) W Rn (ζ ) (X Rn (t))t∈[0,T ] Z

A

( T2n−1 , K T2n−1 , dz) Real part of z ∈ C Reflection of DPP ( , K , λ) First correlation function = density of points m-point correlation function Subset of Rd Set of all permutations of {1, . . . , n} = symmetry group Shift of DPP ( , K , λ) by u ∈ C Signature of permutation σ Special orthogonal group Symplectic group One-dimensional torus = unit circle; T R/2π Z Two-dimensional torus; T2 (R/2π Z) × (R/2π |τ |Z) Theta function with argument z and nome p Four kinds of Jacobi’s theta functions with argument ξ and nome modular parameter τ , j = 0, 1, 2, 3 Unitary group Weyl denominator of type Rn , ζ ∈ Cn Noncolliding Brownian bridges (Bbs) of type Rn with n particles and time duration T Set of all integers {. . . , −1, 0, 1, 2, . . . }

Chapter 1

Introduction

Abstract 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 symbol. The main topic of the present monograph is not the q-extension, but its further extension; the elliptic extension. We introduce a parameter p ∈ C, 0 < | p| < 1 instead of q, which is called a nome, and then define the theta function parameterized by p, θ (z; p), on the complex plane punctured at the origin, z ∈ C× . Basic properties of the theta function used throughout this monograph are summarized here.

1.1 q-Extensions Let R be a collection of all real numbers and C√be of all complex numbers, respectively. The imaginary unit is denoted by i := −1. We fix a parameter q ∈ C so that |q| < 1. For each positive integer n ∈ N := {1, 2, . . . }, the q-analogue [n]q is defined by 1 − qn = 1 + q + · · · + q n−1 . [n]q := 1−q Replacing every occurrence of a positive integer by its q-analogue, the q-analogue of the factorial n! := n · (n − 1) · · · 2 · 1 is given by [n]q ! := [n]q [n − 1]q · · · [2]q [1]q , n ∈ N.

(1.1)

For convenience, we assume[0]q ! = 1. Let N0 := {0, 1, 2, . . . }. Then the q-analogues n n! are defined by of the binomial coefficient = k!(n − k)! k   [n]q ! n , n ∈ N0 , k ∈ {0, 1, . . . , n}. := k q [k]q ![n − k]q ! © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Katori, Elliptic Extensions in Statistical and Stochastic Systems, SpringerBriefs in Mathematical Physics 47, https://doi.org/10.1007/978-981-19-9527-9_1

(1.2)

1

2

1 Introduction

The mathematical symbols (1.1) and (1.2) are called the q-factorial and the q-binomial coefficient, respectively. For n ∈ N0 , k ∈ {0, 1, . . . , n}, the q-analogues of Pascal’s triangle relation are given as       n n−1 n−k n − 1 = +q , k q k−1 q k q       n−1 n n−1 = qk + , k q k q k−1 q in which some powers of q should be inserted at the appropriate places. Such procedures to replace mathematical symbols, identities, functions, and so on by their meaningful q-analogies are generally called q-extensions [6, 21, 51]. The main topic of the present monograph is not the q-extension, but its further extension, which is called an elliptic extension. In order to pass from q-extensions to elliptic extensions smoothly in the next section, here we introduce the q-analogues of the Pochhammer symbol (a)n := a(a + 1) · · · (a + n − 1), n ∈ N, (a)0 := 1 as follows: (a; q)∞ :=

∞ 

(1 − aq j ),

j=0

 (a; q)∞ = (1 − aq j ), n ∈ N, n (aq ; q)∞ j=0 n−1

(a; q)n :=

(1.3)

and (a; q)0 := 1. The q factorials and the q-Pochhammer symbols are related as follows: (q; q)n , n ∈ N0 . [n]q ! = (1 − q)n We notice that the factor (1 − q)n above is irrelevant when the q-binomial coefficient (1.2) is considered:   (q; q)n n = , n ∈ N0 , k ∈ {0, 1, . . . , n}. k q (q; q)k (q; q)n−k The following abbreviation for a product of the q-Pochhammer symbols is used: (a1 , . . . , am ; q)∞ :=

m  (ak ; q)∞ , m ∈ N. k=1

(1.4)

1.2 Theta Functions and Elliptic Extensions

3

By the definition (1.3), the following equalities are readily obtained: (a; q)∞ =

k−1 

(aq ; q )∞ , (a ; q )∞ = j

k

k

k

j=0

k−1 

j

(aωk ; q)∞ , k ∈ N,

(1.5)

j=0

where ωk denotes a primitive k-th root of unity.

1.2 Theta Functions and Elliptic Extensions We write the complex plane which is punctured at the origin as C× := C \ {0} = {z ∈ C : 0 < |z| < ∞}. Let p ∈ C be a fixed number so that 0 < | p| < 1. The theta function with argument z and nome p is defined by1 θ (z; p) := (z, p/z; p)∞ =

∞ 

(1 − zp j )(1 − p j+1 /z).

(1.6)

j=0

We often use the shorthand notation θ (z 1 , . . . , z m ; p) :=

m 

θ (z k ; p), m ∈ N.

k=1

By this definition, we can readily see that lim θ (z; p) = 1 − z.

p→0

It implies that

(1.7)

θ (q n ; p) = [n]q , n ∈ N, p→0 1 − q lim

that is, the p → 0 limit of θ (q n ; p)/(1 − q) is the q-analogue [n]q of a positive integer n ∈ N. Conversely, we may say that θ (q n ; p)/(1 − q) can be regarded as an extension of [n]q by introducing another parameter p in addition to q. Moreover, if we consider θ (aq n ; p) with a = e−2iα and q = e−2iφ , α, φ ∈ [0, 2π ), then (1.7) proves θ (aq n ; p) = sin(α + nφ). lim √ p→0 2i aq n 1

This function is sometimes called the modified theta function. The relations with the four types of Jacobi’s theta functions are given in Sect. 2.4. In this monograph, we will simply call θ(z; p) the theta function.

4

1 Introduction

This suggests that the present extension expressed by theta functions should be regarded as being at the higher level than the trigonometric functions (or the hyperbolic functions). For this reason, this second extension following the q-extension will be called the elliptic extension.2 As a function of z, the theta function θ (z; p) is holomorphic in C× and has only single zeros precisely at p j , j ∈ Z := {0, ±1, ±2, . . . }, that is, the zero set is given by (1.8) {z ∈ C× : θ (z; p) = 0} = { p j : j ∈ Z}. By the definition (1.6) the following equalities are proved: 1 θ (1/z; p) = − θ (z; p), z 1 θ ( pz; p) = − θ (z; p), z

(1.9) (1.10)

which are referred to as the inversion formula and the quasi-periodicity, respectively. (See Exercise 1.1.) By comparing (1.9) and (1.10) using the transformation z → 1/z, we immediately see the symmetry, θ ( p/z; p) = θ (z; p).

(1.11)

Corresponding to (1.5), the following equalities hold: θ (z; p) =

k−1 

θ (zp j ; p k ), θ (z k ; p k ) =

j=0

k−1 

j

θ (zωk ; p), k ∈ N,

(1.12)

j=0

where ωk denotes a primitive k-th root of unity. (See Exercise 1.2.) In C× , θ (z; p) is holomorphic, and it allows a Laurent expansion, θ (z; p) =

 n 1 (−1)n p (2) z n , ( p; p)∞

(1.13)

n∈Z

 where n2 := n(n − 1)/2, n ∈ Z. Since the theta function is defined as the product of two q-Pochhammer symbols as (1.6), the above is written as

Doubly periodic functions which are analytic in any finite domain on C except for poles (i.e., meromorphic) are generally called elliptic functions [57, 78]. Consider θ(eiϕ ; p) by putting p = e2πiτ with Im τ > 0. The periodicity in ϕ → ϕ + 2π is obvious. By (1.11), this has another invariance under ϕ → −ϕ + 2π τ . The theta function is not elliptic, but as suggested by the above observations, elliptic functions can be constructed as rational functions of θ’s. The notion of elliptic extension is much wider and deeper than what we will explain in this monograph, but we can say that the theta function plays important roles everywhere. For advanced studies of elliptic extensions, see, for instance, [2, 4, 7–9, 21, 27, 29, 30, 44, 48, 50, 62, 64, 67, 72–74, 77] and references therein. 2

1.2 Theta Functions and Elliptic Extensions

5

Fig. 1.1 For p = 1/2, the theta function θ(x; p) is plotted for the negative argument x < 0 in the left figure and for the positive argument x > 0 in the right figure, respectively. When x < 0, the theta function is positive definite, but when x > 0 it oscillates around zero infinitely many times as shown by (1.17)

 n (−1)n p (2) z n = ( p; p)∞ (z; p)∞ ( p/z; p)∞ , n∈Z

which is known as Jacobi’s triple product identity. One can show that θ  (1; p) :=

∂θ (z; p)

= −( p; p)2∞ .

z=1 ∂z

(1.14)

The theta function satisfies the Riemann–Weierstrass addition formula [49], u θ (yv, y/v, xu, x/u; p), y (1.15) which is the elliptic extension of the addition formula of trigonometric functions. (See Exercise 1.3.) When p ∈ (0, 1), we see that θ (x y, x/y, uv, u/v; p) − θ (xv, x/v, uy, u/y; p) =

θ (z; p) = θ (z; p).

(1.16)

In this case the definition (1.6) implies the following (see Fig. 1.1): ⎫ θ (x; p) > 0, x ∈ ( p 2 j+1 , p 2 j ) ⎬ θ (x; p) = 0, x = p j ⎭ θ (x; p) < 0, x ∈ ( p 2 j , p 2 j−1 )

j ∈ Z,

θ (x; p) > 0, x ∈ (−∞, 0).

(1.17)

Moreover, we can prove the following. In the interval x ∈ (−∞, 0), θ (x; p) is strictly convex with ∞

 √ min θ (x; p) = θ (− p; p) = (1 + p n−1/2 )2 > 0,

x∈(−∞,0)

n=1

(1.18)

6

1 Introduction

Fig. 1.2 For p = 1/4, the theta function θ(x; p) is plotted for x < 0. It is strictly√convex and at x = − 1/4 = −0.5 it attains its minimum which is positive

Fig. 1.3 For p = 1/4, the theta function θ(x; p) is plotted for x ∈ ( p, 1) = (0.25, 1). It is strictly concave √ and at x = 1/4 = 0.5 it attains its maximum. At both edges of the interval, x = p = 0.25 and x = 1, θ(x) becomes zero

and lim x↓−∞ θ (x; p) = lim x↑0 θ (x; p) = +∞, and in the interval x ∈ ( p, 1), θ (x; p) is strictly concave with ∞

 √ max θ (x; p) = θ ( p; p) = (1 − p n−1/2 )2 ,

x∈( p,1)

(1.19)

n=1

θ (x; p) ∼ ( p; p)2∞ (x − p)/ p as x ↓ p, and θ (x; p) ∼ ( p; p)2∞ (1 − x) as x ↑ 1, where (1.10) and (1.14) were used. See Figs. 1.2 and 1.3.

Exercises 1.1 By the definition of the theta function (1.6), prove the inversion formula (1.9) and the quasi-periodicity (1.10). 1.2 By direct calculation, verify the following equalities: θ (ζ ; p 3 )θ (ζ p; p 3 )θ (ζ p 2 ; p 3 ) = θ (ζ ; p), θ (ζ ;

p)θ (ζ ω3 ; p)θ (ζ ω32 ;

where ω3 is a primitive 3rd root of unity.

p) = θ (ζ ; p ), 3

3

(1.20) (1.21)

1.2 Theta Functions and Elliptic Extensions

7

1.3 Let x = e−2ia , y = e−2ib , u = e−2ic , and v = e−2id in the Riemann–Weierstrass addition formula (1.15). Consider the limit p → 0 and derive the following trigonometric equality: sin(a + b) sin(a − b) sin(c + d) sin(c − d) − sin(a + d) sin(a − d) sin(c + b) sin(c − b) = sin(b + d) sin(b − d) sin(a + c) sin(a − c).

(1.22)

Show that (1.22) is readily verified if we use the following addition formulas of trigonometric functions: sin(a ± b) = sin a cos b ± cos a sin b.

(1.23)

Chapter 2

Brownian Motion and Theta Functions

Abstract We introduce the Brownian motion on a real line R. First we notice that its transition probability density solves the heat equation starting from a single delta function. Then we consider the Brownian motion on a unit circle, which is regarded as a one-dimensional torus and is denoted by T. Two different formulas of the transition probability are given, both of which are expressed using the theta function with different nomes. The equivalence of these two expressions implies Jacobi’s imaginary transformation of the theta function. We also study the Brownian motion on a semicircle which is identified with the interval [0, π ] with two boundary points 0 and π . We impose the absorbing boundary condition or the reflecting boundary condition at each of the boundary points and hence we obtain four types of Brownian motion in the interval. We see an interesting correspondence between these four types of Brownian motion and the four types of Jacobi’s theta functions via expressions of the transition probability densities.

2.1 Brownian Motion on R The one-dimensional real space is denoted by R. We consider the motion of a Brownian particle in R starting from the origin 0 at time t = 0. Each realization of its random trajectory is called a sample path, denoted by ω. Let  be the collection of all sample paths and call it the sample path space. In a path ω ∈  the position of the Brownian particle at each time t ≥ 0 is written as B(t, ω). Usually we omit ω and simply write it as B(t), t ≥ 0. Each event associated with the process is represented by a subset of , and the collection of all events is denoted by F . The whole sample path space  and the empty set ∅ are in F . For any two sets A, B ∈ F , we assume that A ∪ B ∈ F and A ∩ B ∈ F . If A ∈ F , then its complement Ac :=  \ A is also in F . It is closed for any infinite sum of events in the sense that, if An ∈ F , n = 1, 2, . . . , then ∪n≥1 An ∈ F . Such a collection is said to be a σ -field. A probability measure P is a function defined on the σ -field F . Since any element of F is given by a set as above, any input of P is a set, and hence P is a set function. It satisfies the following properties; P[A] ≥ 0 for all A ∈ F , © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Katori, Elliptic Extensions in Statistical and Stochastic Systems, SpringerBriefs in Mathematical Physics 47, https://doi.org/10.1007/978-981-19-9527-9_2

9

10

2 Brownian Motion and Theta Functions

P[] =1, P[∅] = 0,and if An ∈ F , n ∈ N, are disjoint; An ∩ Am = ∅, n = m, ∞ c then P[ ∞ n=1 An ] = n=1 P[An ]. In particular, P[A ] = 1 − P[A] for all A ∈ F . The triplet (, F , P) is called theprobability space. The smallest σ -field containing all intervals on R is called the Borel σ -field and denoted by B(R). A random variable is a measurable function f (ω) on  such that, for every Borel set A ∈ B(R), f −1 (A) ∈ F . Two events A and B are said to be independent if P[A ∩ B] = P[A]P[B]. Two random variables X and Y are independent if the events A = {X : X ∈ A} and B = {Y : Y ∈ B} are independent for any A, B ∈ B(R). The one-dimensional standard Brownian motion, {B(t, ω) : t ≥ 0}, has the following three properties: (BM1) P[B(0, ω) = 0] = 1. (BM2) There is a subset  ⊂ , such that P[] = 1 and for any ω ∈ , B(t, ω) is a real continuous function of t. We say that B(t) has a continuous path almost surely (a.s., for short). (BM3) For an arbitrary n ∈ N := {1, 2, 3, . . . }, and for any sequence of times, t0 := 0 < t1 < · · · < tn , the increments B(tm ) − B(tm−1 ), m = 1, 2, . . . , n, are independent, and each increment is in the normal distribution (Gaussian distribution) with mean 0 and variance σ 2 = tm − tm−1 . It means that for any 1 ≤ m ≤ M and a < b,  b

P[B(tm ) − B(tm−1 ) ∈ [a, b]] =

p(tm−1 , 0; tm , x)d x,

a

where for 0 ≤ s ≤ t < ∞, x, y ∈ R we define ⎧ 2 ⎨√ 1 e−(y−x) /2(t−s) , p(s, x; t, y) := 2π(t − s) ⎩ δ(x − y),

for t > s,

(2.1)

for t = s.

It is obvious from (2.1) in the property (BM3) that p(s, x; t, y) = p(0, 0; t − s, y − x), p(s, x; t, y) = p(s, y; t, x), 0 ≤ s ≤ t, x, y ∈ R.

(2.2)

By (BM3) we see that, for any 0 ≤ s < t < ∞, x ∈ R, and A ∈ B(R),  P[B(t) ∈ A|B(s) = x] =

p(s, x; t, y)dy, A

where the LHS denotes the probability for B(t) ∈ A with the condition that the position of the one-dimensional standard Brownian motion was x at time s. The function p(s, x; t, y) is then called the transition probability density of the

2.1 Brownian Motion on R

11

one-dimensional standard Brownian motion.1 It gives the fundamental solution of the heat equation (diffusion equation) in the sense that 1 ∂2 ∂ p(0, x; t, y) = p(0, x; t, y) ∂t 2 ∂ y2 1 ∂2 = p(0, x; t, y), 2 ∂x2

(2.3)

with lim p(0, x; t, y) = δ(x − y). t↓0

(2.4)

Moreover, it is easy to verify that the Chapman–Kolmogorov equation holds:  R

p(s, x; t, y)p(t, y; u, z)dy = p(s, x; u, z), 0 ≤ s ≤ t ≤ u, x, z ∈ R, (2.5)

which implies the Markov property of Brownian motion. For any c > 0, the following equality is satisfied by the transition probability (2.1) of the one-dimensional standard Brownian motion: p(c2 s, cx; c2 t, cy)d(cy) = p(s, x; t, y)dy.

(2.6)

This implies the equivalence in probability law (law)

(B(t))t≥0 =



1 B(c2 t) c

, ∀c > 0,

(2.7)

t≥0

which is called Brownian motion scaling. It is obvious that if we perform a time change t → 2αt, α > 0, the obtained transition probability density p(0, x; 2αt, y) = √

1 4π αt

e−(y−x)

2

/4αt

, t >0

satisfies the heat equation with diffusivity constant α, ∂2 ∂ p(0, x; 2αt, y) = α 2 p(0, x; 2αt, y). ∂t ∂y The standard Brownian motion is a special case with α = 1/2.

1

By the shift invariance in space and time given by the first equality in (2.2), the property (BM1) is readily generalized: The Brownian motion started at an arbitrary fixed point x ∈ R a.s. is defined by B x (t) := x + B(t), t ≥ 0. The transition probability density for B x (t) is given by p(0, x; t, y), t ≥ 0, y ∈ R.

12

2 Brownian Motion and Theta Functions

From now on, we refer to the one-dimensional standard Brownian motion simply as Brownian motion.

2.2 Brownian Motion on the One-Dimensional Torus T We write the unit circle on the complex plane as T := {z ∈ C : |z| = 1}. It is a one-dimensional torus; T = R/2π Z. Each point in T is expressed by ei x , x ∈ [0, 2π ). In the following we will introduce a Brownian motion on T. The transition probability density is denoted by pT (s, x; t, y) for 0 < s < t < ∞ and x, y ∈ [0, 2π ). This should solve the heat equation (2.3) with (2.4), and satisfy the periodic condition pT (s, x; t, y + 2π n) = pT (s, x; t, y), ∀n ∈ Z. For this periodicity, we assume that it is expressed in the form pT (0, x; t, y) =

∞ 

Cn (t) cos(n(y − x))

n=−∞

with a series of time-dependent coefficients (Cn (t))n∈Z , t ≥ 0. In order to solve (2.3), 2 the time dependence of Cn (t) is determined as Cn (t) = cn e−n t/2 . Then the initial condition (2.4) determines cn ≡ 1/2π for all n ∈ Z, since δ(x) is expressed as δ(x) =

∞ ∞ 1  inx 1  e = cos(nx), x ∈ R. 2π n=−∞ 2π n=−∞

On the other hand, pT will be obtained by ‘wrapping’ the transition probability density on R given by (2.1), pT (0, x; t, y) =

∞ 

p(0, x; t, y + 2π w),

w=−∞

where w denotes the winding number of the Brownian path around the circle. Hence we have two different expressions for pT :

2.2 Brownian Motion on the One-Dimensional Torus T

pT (0, x; t, y) = =

13

∞ 1  −n 2 t/2 e cos(n(y − x)) 2π n=−∞ ∞  w=−∞

√

1 2π t

e−(y−x+2πw)

2

/2t

, t > 0, x, y ∈ [0, 2π ).

Compare with the Laurent expansion of the theta function (1.13); the above two expressions are written using the theta functions with different arguments and nomes: ( p t ; p t )∞ 1/2 θ (− pt ei(y−x) ; pt ) 2π 2 e−(y−x) /2t 1/2 pt )∞ θ (− pt e−2π(y−x)/t ;  pt ), x, y ∈ [0, 2π ), = √ ( pt ;  2π t (2.8)

pT (0, x; t, y) =

where

pt := e−4π pt := e−t , 

2

/t

, t > 0.

(2.9)

The above provides a probability theoretical proof of the equality between the theta functions [1], which is known as Jacobi’s imaginary transformation. The argument 1/2 1/2 pt e−2π(y−x)/t in the − pt ei(y−x) in the former expression is transformed to − latter one in (2.8), and hence the coordinates x and y in T seem to be changed to pure imaginaries 2πi x/t and 2πi y/t, respectively. Here we give a general form of Jacobi’s imaginary transformation as a lemma. For the nome p ∈ C, | p| < 1, we define the nome modular parameter τ by p = e2πiτ =: p(τ ).

(2.10)

By definition, τ ∈ C and Im τ > 0. Lemma 2.1 With (2.10), we define  p := p(−1/τ ) = e−2πi/τ .

(2.11)

Then, for ζ ∈ C, the following equality holds: θ (eiζ ; p) =

1  ζ  −iζ /τ p 1/8 ( p;  p )∞ e3πi/4  ζ2 + i 1 − θ (e exp − i ; p ). τ 1/2 p 1/8 ( p; p)∞ 4π τ τ 2 (2.12)

By the above construction, the symmetry (2.2) of the transition probability p of the Brownian motion on R is inherited as

14

2 Brownian Motion and Theta Functions

pT (s, x; t, y) = pT (0, 0; t − s, y − x), pT (s, x; t, y) = pT (s, y; t, x), for any 0 ≤ s ≤ t, x, y ∈ [0, 2π ).

(2.13)

The second equality in the above can be directly proved by applying the symmetry 1/2 of the theta function (1.11) to the first expression in (2.8) as θ (− pt ei(y−x) ; pt ) = 1/2 i(y−x) 1/2 i(x−y) θ ( pt /(− pt e ); pt ) = θ (− pt e ; pt ). We see a reciprocity of time in (2.9). By the asymptotic (1.7) of the theta function, we see that the former expression is useful to estimate the long-term behavior of the transition probability as 1 1 1/2 (1 + pt ei(y−x) ) ∼ as t → ∞, 2π 2π

pT (0, x; t, y) ∼

since pt → 0 as t → ∞, while the latter is useful for the short-term behavior as e−(y−x) /2t 1/2 pT (0, x; t, y) ∼ √ (1 +  pt e−2π(y−x)/t ) ∼ δ(y − x) as t → 0, 2π t 2

since  pt → 0 as t → 0. The above shows that the distribution of the location of B(t) started from a delta measure at x is relaxed to the uniform measure on T as t → ∞. The Chapman–Kolmogorov equation (2.5) for p is expressed for pT as  T

pT (s, x; t, y)pT (t, y; u, z)dy = pT (s, x; u, z), 0 ≤ s < t < u, x, z ∈ [0, 2π ), (2.14)

and it proves the convolution formula for the theta function, 1 2π



2π

0

=

1/2

1/2

θ (− pt−s ei(y−x) ; pt−s )θ (− pu−t ei(z−y) ; pu−t )dy

( pu−s ; pu−s )∞ 1/2 θ (− pu−s ei(z−x) ; pu−s ), ( pt−s ; pt−s )∞ ( pu−t ; pu−t )∞

(2.15)

0 ≤ s < t < u, x, z ∈ [0, 2π ). Here notice the multiplicities in arguments and nomes 1/2 1/2 1/2 such that pt−s ei(y−x) pu−t ei(z−y) = pu−s ei(z−x) and pt−s pu−t = pu−s , which are guaranteed by our choice (2.9) and are needed to establish (2.15).

2.3 Brownian Motion in the Interval [0, π] Now we consider the upper-half circle, T ∩ {z ∈ C : Im z ≥ 0} and identify it with the interval [0, π ] in R. We consider a Brownian motion in [0, π ] imposing one of the two kinds of boundary conditions. One of them is the absorbing boundary condition for Brownian motion such that, at the boundary point b ∈ {0, π }, the Brownian motion is killed. The corresponding transition probability density should satisfy the Dirichlet

2.3 Brownian Motion in the Interval [0, π ]

15

boundary condition. Another one is the reflecting boundary condition such that, at b ∈ {0, π }, the transition probability density should satisfy the Neumann boundary condition.

2.3.1 Absorbing at Both Boundary Points The transition probability density denoted by paa [0,π] (0, x; t, y) should solve the heat equation 1 ∂2 ∂ u(t, y) = u(t, y), t > 0 (2.16) ∂t 2 ∂ y2 with the initial condition u(0, y) = δ(x − y), x, y ∈ [0, π ],

(2.17)

under the Dirichlet boundary condition u(t, 0) = u(t, π ) = 0, t ≥ 0.

(2.18)

For (2.18), we choose the basis of series expansion as {sin(ny)}n∈Z and obtain the result, paa [0,π] (0, x; t,

∞ 1  −n 2 t/2 y) = e sin(nx) sin(ny), t ≥ 0, x, y ∈ [0, π ]. (2.19) π n=−∞

It is easy to confirm that this is written as follows: paa [0,π] (0, x; t, y) =

∞ 

{p(0, x; t, y + 2π w) − p(0, −x; t, y + 2π w)}

w=−∞

= pT (0, x; t, y) − pT (0, −x; t, y) ( p t ; p t )∞ 1/2 1/2 = {θ (− pt ei(y−x) ; pt ) − θ (− pt ei(y+x) ; pt )}, (2.20) 2π t > 0, x, y ∈ [0, π ]. The expression (2.19) is called the spectral representation of the transition probability density in Appendix 1.6 of [3]. The first expression in (2.20) coincides with the formula given there which is obtained by applying the reflection principle of Brownian motion.

16

2 Brownian Motion and Theta Functions

2.3.2 Reflecting at Both Boundary Points The transition probability density denoted by prr[0,π] (0, x; t, y) should solve (2.16) with the initial condition (2.17) under the Neumann boundary condition   ∂u(t, y)  ∂u(t, y)  = = 0, t ≥ 0. ∂ y  y=0 ∂ y  y=π

(2.21)

For (2.21), we choose the basis of series expansion as {cos(ny)}n∈Z and obtain the result, ∞ 1  −n 2 t/2 e cos(nx) cos(ny), t ≥ 0, x, y ∈ [0, π ]. π n=−∞ (2.22) It is easy to confirm that this is written as follows:

prr[0,π] (0, x; t, y) =

prr[0,π] (0, x; t, y) =

∞ 

{p(0, x; t, y + 2π w) + p(0, −x; t, y + 2π w)}

w=−∞

= pT (0, x; t, y) + pT (0, −x; t, y) ( p t ; p t )∞ 1/2 1/2 = {θ (− pt ei(y−x) ; pt ) + θ (− pt ei(y+x) ; pt )}, (2.23) 2π t > 0, x, y ∈ [0, π ]. The former expression coincide with the formula given in Appendix 1.5 in [3], where the expression (2.22) is the spectral representation of the transition probability density.

2.3.3 Absorbing at One Boundary Point and Reflecting at Another Boundary Point The transition probability density denoted by par [0,π] (0, x; t, y) should solve (2.16) with the initial condition (2.17) under the following boundary condition: u(t, 0) = 0,

 ∂u(t, y)  = 0, t ≥ 0. ∂ y  y=π

(2.24)

For (2.24), we choose the basis of series expansion as {sin{(n − 1/2)y}}n∈Z and obtain the result, par [0,π] (0, x; t, y) =

∞ 1  −(n−1/2)2 t/2 e sin{(n − 1/2)x} sin{(n − 1/2)y}, (2.25) π n=−∞

2.3 Brownian Motion in the Interval [0, π ]

17

t ≥ 0, x, y ∈ [0, π ]. This is written as follows [38]: 1/8

p t ( p t ; p t )∞ 2π × {e−i(y−x)/2 θ (−ei(y−x) ; pt ) − e−i(y+x)/2 θ (−ei(y+x) ; pt )},

par [0,π] (0, x; t, y) =

(2.26)

t > 0, x, y ∈ [0, π ]. By changing the variables as x → π − x and y → π − x, we can exchange the ar boundary conditions at 0 and π , and hence pra [0,π ] (0, x; t, y) = p[0,π ] (0, π − x; t, π − y). We have the following results: pra [0,π] (0, x; t, y) =

∞ 1  −(n−1/2)2 t/2 e cos{(n − 1/2)x} cos{(n − 1/2)y} π n=−∞

1/8

p t ( p t ; p t )∞ 2π −i(y−x)/2 θ (−ei(y−x) ; pt ) + e−i(y+x)/2 θ (−ei(y+x) ; pt )}, × {e

=

(2.27)

t > 0, x, y ∈ [0, π ]. We introduce the following sum of the transition probability densities on R with alternating signs: ∞ 

 pT (0, x; t, y) :=

(−1)w p(0, x; t, y + 2π w).

(2.28)

w=−∞

Using Jacobi’s imaginary transformation (2.9), we can rewrite the above as 1/8

 pT (0, x; t, y) =

pt ( pt ; pt )∞ −i(y−x)/2 e θ (−ei(y−x) ; pt ). 2π

(2.29)

Hence (2.26) and (2.27) are expressed as follows: par pT (0, x; t, y) −  pT (0, −x; t, y), [0,π] (0, x; t, y) =  pra pT (0, x; t, y) +  pT (0, −x; t, y), t > 0, x, y ∈ [0, π ]. [0,π] (0, x; t, y) =  (2.30) We notice that  pT satisfies the Chapman–Kolmogorov equation,  T

 pT (s, x; t, y) pT (t, y; u, z)dy =  pT (s, x; u, z), 0 ≤ s < t < u, x, z ∈ [0, 2π ),

(2.31) and p [0,π] ’s with ∈ {aa, ar, ra, rr} also do. It is easy to verify the following symmetry:

18

2 Brownian Motion and Theta Functions







p[0,π] (s, x; t, y) = p[0,π] (0, 0; t − s, y − x), p[0,π] (s, x; t, y) = p[0,π] (s, y; t, x), for any 0 < s < t, x, y ∈ [0, π ].

(2.32)

The former is obvious by the above definition. The latter is directly shown using the symmetry (1.11) of the theta function to the expressions (2.20) and (2.23) for rr ar paa [0,π] and p[0,π] , and the inversion formula (1.9) to (2.26) and (2.27) for p[0,π] ra −i(y−x)/2 i(y−x) −i(y−x)/2 i(y−x) i(y−x) and p[0,π] , as e θ (−e ; pt ) = e e θ (1/(−e ); pt ) = e−i(x−y)/2 θ (−ei(x−y) ; pt ).

2.4 Expressions by Jacobi’s Theta Functions Here we introduce the four kinds of Jacobi’s theta functions denoted by ϑ j , j = 0, 1, 2, 3, since they are useful to express the results given above. We use the nome modular parameter τ defined by (2.10). Using the theta function θ (z; p) defined by (1.6), ϑ1 is defined as follows [57, 78]: ϑ1 (ξ ; τ ) := i p 1/8 ( p; p)∞ e−iξ θ (e2iξ ; p) ∞ 

=i =2

(−1)n p (n−1/2)

2

n=−∞ ∞ 

/2 (2n−1)iξ

(−1)n−1 e(n−1/2)

2

e

πiτ

sin{(2n − 1)ξ }.

(2.33)

n=1

Then other three kinds of Jacobi’s theta functions are defined as [57, 78] ϑ0 (ξ ; τ ) := −iei(ξ +πτ/4) ϑ1 (ξ + π τ/2; τ ) = ( p; p)∞ θ ( p 1/2 e2iξ ; p) =

∞ 

(−1)n p n

2

/2 2πiξ

e

=1+2

n=−∞

ϑ2 (ξ ; τ ) := ϑ1 (ξ + π/2; τ ) = p 1/8 ( p; =

∞ 

p (n−1/2)

2

/2 (2n−1)iξ

e

∞  2 (−1)n en πiτ cos(2nξ ),

n=1 p)∞ e−iξ θ (−e2iξ ; ∞  (n−1/2)2 πiτ

=2

n=−∞

e

(2.34)

p)

cos{(2n − 1)ξ }, (2.35)

n=1

ϑ3 (ξ ; τ ) := ei(ξ +πτ/4) ϑ1 (ξ + π(1 + τ )/2; τ ) = ( p; p)∞ θ (− p 1/2 e2iξ ; p) =

∞  n=−∞

pn

2

/2 2πiξ

e

=1+2

∞ 

en

2

πiτ

cos(2nξ ).

(2.36)

n=1

Then the transition probability densities of the Brownian motions on T or in [0, π ] given in the previous section are expressed as follows [38]: For t > 0,

2.4 Expressions by Jacobi’s Theta Functions

1 ϑ3 ((y − x)/2; it/2π ) 2π = p(0, x; t, y)ϑ3 (iπ(y − x)/t; 2πi/t), x, y ∈ [0, 2π ),

19

pT (0, x; t, y) =

(2.37)

and  pT (0, x; t, y) = paa [0,π] (0, x; t, y) = prr[0,π] (0, x; t, y) = par [0,π] (0, x; t, y) = pra [0,π] (0, x; t, y) =

1 ϑ2 ((y − x)/2; it/2π ), x, y ∈ [0, 2π ), 2π 1 {ϑ3 ((y − x)/2; it/2π ) − ϑ3 ((y + x)/2; it/2π )}, 2π 1 {ϑ3 ((y − x)/2; it/2π ) + ϑ3 ((y + x)/2; it/2π )}, 2π 1 {ϑ2 ((y − x)/2; it/2π ) − ϑ2 ((y + x)/2; it/2π )}, 2π 1 {ϑ2 ((y − x)/2; it/2π ) + ϑ2 ((y + x)/2; it/2π )}, 2π x, y ∈ [0, π ]. (2.38)

Exercise 2.1 The two expressions of pT using Jacobi’s theta function ϑ3 in (2.37) correspond to the two expressions in (2.8), respectively. The nome modular parameter is it/2π for the first expression, while it is 2πi/t for the second one, and they are related by Jacobi’s imaginary transformation. Give the expressions for other transition probability densities in (2.38) using suitable Jacobi’s theta functions with the nome modular parameter 2πi/t.

Chapter 3

Biorthogonal Systems of Theta Functions and Macdonald Denominators

Abstract The fact lim p→0 θ (ζ ; p) = 1 − ζ , ζ ∈ C× suggests that the theta function θ (ζ ; p) is an elliptic analogue of a linear function of ζ . What is the elliptic analogue of a polynomial of ζ ? Rosengren and Schlosser gave seven kinds of answers to this fundamental question by introducing seven infinite series of spaces of theta functions associated with the irreducible reduced affine root systems, Rn = An−1 , Bn , Bn∨ , Cn , Cn∨ , BCn , n ∈ N, and Dn , n ∈ {2, 3, . . . }. Here n indicates the degree of the elliptic analogues of polynomials. The basis functions for the function spaces are called the Rn theta functions and are denoted by {ψ jRn }nj=1 . It was proved that the determinants consisting of {ψ jRn }nj=1 provide the Macdonald denominator formulas, which are the elliptic extensions of the Weyl denominator formulas. In this chapter, first we give a brief review of the theory of Rosengren and Schlosser. Then we introduce appropriate inner products and prove the biorthogonality relations for the Rn theta functions of Rosengren and Schlosser.

3.1

An−1 Theta Functions and Determinantal Identity

The elliptic analogue of a polynomials of θ might be given by a product of θ ’s. We have to remark on the equalities (1.12), however, since they show that a degree of product of θ ’s depends on a choice of nome. In order to define a degree of polynomials of θ ’s with respect to a specified nome, Rosengren and Schlosser [64] generalized the notion of the quasi-periodicity (1.10) as follows. Definition 3.1 Assume that f (ζ ) is holomorphic in C× and n ∈ N. If there is a parameter r ∈ R \ {0} and f satisfies the equality f ( pζ ) =

(−1)n f (ζ ), rζn

then f is said to be an An−1 theta function of norm r . The space of all An−1 theta A functions with nome p and norm r is denoted by E p,rn−1 . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Katori, Elliptic Extensions in Statistical and Stochastic Systems, SpringerBriefs in Mathematical Physics 47, https://doi.org/10.1007/978-981-19-9527-9_3

21

22

3 Biorthogonal Systems of Theta Functions and Macdonald Denominators

By this definition, we can say that θ (ζ ; p) is an A0 theta function of norm r = 1. The following is proved [74]. (See Exercise 3.1.) A

A

Lemma 3.1 The space E p,rn−1 is n-dimensional and {ψ j n−1 (ζ ; p, r )}nj=1 defined by A

ψ j n−1 (ζ ; p, r ) := ζ j−1 θ ( p j−1 (−1)n−1 r ζ n ; p n ) = ζ j−1

n−1 

θ (α j−1 βωnk ζ ; p),

j = 1, . . . , n,

(3.1)

k=0

form a basis, where α and β are complex numbers such that α n = p, β n = (−1)n−1 r , respectively, and ωn is a primitive n-th root of unity. By (1.7), we see that A

A

ψ j n−1 (ζ ; 0, r ) := lim ψ j n−1 (ζ ; p, r ) p→0  1 − (−1)n−1 r ζ n , = ζ j−1 ,

j = 1, j = 2, . . . , n.

A

Hence E0,rn−1 spanned by them is a space of polynomials of degree n in the form c0 + c1 ζ + · · · + cn ζ n with a constraint such that the ratio cn /c0 is fixed to be −(−1)n−1 r . A It implies that dim E0,rn−1 = n. It is easy to perform the following calculation:  det

1≤ j,k≤n

 A ψ j n−1 (ζk ; 0, r ) = =

det

1≤ j,k≤n

  j−1 ζk − (−1)n−1 r ζkn δ j1

n    (−1)1+ 1 − (−1)n−1 r ζ n =1

det

1≤ j≤n−1 1≤k≤n,k=

  j ζk

    ( j−1)+nδ j1 j−1 = det ζk − r (−1)n−1 det ζk 1≤ j,k≤n 1≤ j,k≤n

n    j−1 = 1−r . ζ det ζk =1

1≤ j,k≤n

Hence we have the equality

n    An−1 det ψ j (ζk ; 0, r ) = 1 − r ζ W An−1 (ζ ),

1≤ j,k≤n

(3.2)

=1

for ζ = (ζ1 , . . . , ζn ) ∈ Cn , where W An−1 (ζ ) :=

 1≤ j 0, we consider the interval [0, π] representing the half perimeter of the circle T and the Weyl alcove, Wn ([0, π]) := {x ∈ Rn : 0 ≤ x1 < · · · < xn ≤ π}. We choose the following equidistance configurations of n particles in Wn ([0, π]): u = u Rn = (u 1Rn , . . . , u nRn ) ⎧ ⎪ j − 1/2, 2π ⎨ Rn with u j := × j, ⎪ N ⎩ j − 1,

for Rn = Bn , Bn∨ , for Rn = Cn , Cn∨ , BCn , for Rn = Dn ,

(4.24)

where N = N Rn given by (3.9). See Fig. 4.2. Then we define { Rjkn ( pt )}1≤ j,k≤n as follows. For j, k = 1, . . . , n, 4π −γ 2j /2 1 pt  ck ( p N 2 t ; p N 2 t )∞ N  i sin(γ j u k ), for Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , × for Rn = Dn , cos(γ j u k ), (4.25)

 jk ( pt ) =  Rjkn ( pt ) :=

4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π]

0

π

47

2π

An−1 Bn Bn Cn Cn BCn Dn Fig. 4.2 The equidistance configurations u Rn are shown when n = 5. They are used for the initial and final configurations of the noncolliding Brownian bridges. The boundary conditions are summarized by (4.35). Since the boundary condition is (aa) for Bn∨ and Cn , there should be no point both at 0 and π, while for Dn we can put points both at 0 and π, since the boundary condition is (rr). For Bn , Cn∨ , BCn , the origin should be vacant, since the boundary condition (a) is imposed at 0

where ⎧ ⎪ Rn = Bn , Cn∨ , BCn , ⎨ j − n − 1/2, Rn γ j = γ j := j − n − 1, Rn = Bn∨ , Cn , ⎪ ⎩ j − n, R n = Dn , ⎧ 1, k = 1, . . . , n, Rn = Bn∨ , Cn , BCn , ⎪ ⎪  ⎪ ⎪ ⎪ k = 1, . . . , n − 1, ⎪ ⎨ 1, Rn = Bn , Cn∨ , Rn 1/2, k = n,  ck =  ck :=  ⎪ ⎪ ⎪ k = 2, . . . , n − 1, ⎪ 1, ⎪ ⎪ R n = Dn , ⎩ 1/2, k = 1, n, N = N Rn given by (3.9), and u = u Rn given by (4.24). We can prove the following [38].

48

4 KMLGV Determinants and Noncolliding Brownian Bridges

Lemma 4.4 For j = 1, . . . , n, t > 0, and x ∈ [0, π], n 

−i x/2 Rn i x  j ( pt )par ψ j (e ; pNt ), if Rn = Bn , Cn∨ , BCn , [0,π] (0, u  ; t, x) = e

=1 n 

Rn i x ∨  j ( pt )paa [0,π] (0, u  ; t, x) = ψ j (e ; pNt ), if Rn = Bn , C n ,

=1 n 

 j ( pt )prr[0,π] (0, u  ; t, x) = ψ Dj n (ei x ; p(2n−2)t ), if Rn = Dn ,

(4.26)

=1

where  j ( pt ) =  Rjn ( pt ) given by (4.25) and u = u Rn given by (4.24). Proof (i) First we prove (4.26) for Rn = Cn and Bn∨ . By (2.20), (4.24), and (4.25), in these cases we have L Rj n :=

n 

 Rjn ( pt )paa [0,π] (0, u  ; t, x)

=1 −γ 2 /2

2i( pt ; pt )∞ pt j = (p 2 ; p 2 ) N   Nn t N t ∞ n   1/2 i(x−u  ) 1/2 i(x+u  ) sin(γ j u  )θ(− pt e ; pt ) − sin(γ j u  )θ(− pt e ; pt ) . × =1

=1

(4.27) Consider the case that Rn = Cn . Since sin(γ Cn u C n ) = sin(γ Cj n π/(n + 1)) = 0 if  = 0 and  = n + 1 for γ Cj n = j − n − 1 ∈ Z, we can change the range of  in the first summation in the bracket from {1, . . . , n} to {0, 1, . . . , n, n + 1}. In the second summation there, we replace the variable  by − and change the range from {1, . . . , n} to {−n, −n + 1, . . . , −1}. Then the above is written as L Cj n

−γ 2 /2 n+1 

2i( pt ; pt )∞ pt j = ( p N 2 t ; p N 2 t )∞ N

sin(γ j u  )θ(− pt ei(x−u  ) ; pt ), 1/2

=−n

where N = N Cn . The range of summation is now {−n, . . . , n + 1} which consists of 2n + 2 = N Cn elements. By (1.13), ( pt ; pt )∞ θ(− pt ei(x−u  ) ; pt ) = 1/2

 m∈Z

ptm ei(x−u  )m . 2

4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π]

49

Then we have −γ /2  m 2 /2 pt j = pt ei xm ( p N 2 t ; p N 2 t )∞ m∈Z   n+1 n+1 1  i(γ−m)u  1  i(−γ−m)u  × e − e . N =−n N =−n 2

L Cj n

We can use the identity (4.5) to obtain the equalities n+1  1  i(±γ−m)u   e = 1(m = ±γ + Nk) = 1(m = ±(γ + Nk)), N =−n k∈Z

k∈Z

and the above gives −γ 2 /2

L Cj n

pt j = (p 2 ; p 2 ) Nt Nt ∞   (γ j +Nk)2 /2  (γ j +Nk)2 /2 i(γ j +Nk)x −i(γ j +Nk)x × pt e − pt e . k∈Z

k∈Z

We see that, for γ j = γ Cj n = j − n − 1 and N = N Cn = 2n + 2,   1 k 1 . (γ j + Nk)2 = γ 2j + N jk + N 2 2 2 2 Hence we obtain −γ 2 /2

L Cj n

pt j = ( p N 2 t ; p N 2 t )∞

 γ 2 /2

pt j eiγx



N (k ) j pNt 2 ( pNt eiN x )k

k∈Z γ 2j /2

− pt

e

−iγx



N (k ) j pNt 2 ( pNt e−iN x )k



k∈Z

=e

iγx

j θ(− pNt eiN x ;

N pNt )

N − e−iγx θ(− pNt e−iN x ; pNt ). j

Since γ Cj n = j − n − 1 = αCj n given by (3.15), (4.26) is proved for Rn = Cn . Next consider the case that Rn = Bn∨ . In the second summation in the bracket of (4.27), we change the variable as − + 1 → . Then (4.27) is written as B∨ Ljn

−γ 2 /2

2i( pt ; pt )∞ pt j = ( p N 2 t ; p N 2 t )∞ N

n  =−n+1

sin(γ j u  )θ(− pt ei(x−u  ) ; pt ), 1/2

50

4 KMLGV Determinants and Noncolliding Brownian Bridges ∨

where the range of summation is {−n + 1, . . . , n} consisting of 2n = N Bn elements. The above is written as −γ /2  m 2 /2 pt j = pt ei xm ( p N 2 t ; p N 2 t )∞ m∈Z  n 1  i(γ−m)2π(−1/2)/N Bn∨ 1 × e − N =−n+1 N 2

B∨ Ljn

n 

 e

∨

i(−γ−m)2π(−1/2)/N Bn

.

=−n+1

We use the identity (4.5) to obtain the equalities n+1 1  i(±γ−m)2π(−1/2)/N  e = (−1)k 1(m = ±(γ + Nk)) N =−n k∈Z ∨

with N = N Bn = 2n, and then we have −γ 2 /2

pt j = ( p N 2 t ; p N 2 t )∞     2 2 (γ +Nk) /2 (γ +Nk) /2 j j × (−1)k pt ei(γ j +Nk)x − (−1)k pt e−i(γ j +Nk)x .

B∨ Ljn

k∈Z

k∈Z

Following the calculation shown above for Cn , (4.26) is also proved for Rn = Bn∨ . (ii) Next we prove (4.26) for Rn = BCn , Bn , and Cn∨ . By (2.26), (4.24), and (4.25), in these cases we have L Rj n :=

n 

 Rjn ( pt )par [0,π] (0, u  ; t, x)

=1 −γ 2 /2

1/8

2i pt ( pt ; pt )∞ pt j = ( p N 2 t ; p N 2 t )∞ N −

n 



n 

 c sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt )

=1

 c sin(γ j u  )e

−i(x+u  )/2

 θ(−e

i(x+u  )

; pt ) .

(4.28)

=1

Consider the case that Rn = BCn . Since sin(γu BCn ) = sin(γ2π/(2n + 1)) = 0 if  = 0, we can change the range of  in the first summation in the bracket from {1, . . . , n} to {0, 1, . . . n}. In the second summation there, we replace the variable  by − and change the range from {1, . . . , n} to {−n, −n + 1, . . . , −1}. Since  cBCn = 1 ∀, the above is written as

4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π] −γ 2 /2

1/8

n L BC j

2i pt ( pt ; pt )∞ pt j = ( p N 2 t ; p N 2 t )∞ N

n 

51

sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ), (4.29)

=−n

where the range of summation is {−n, . . . , n} consisting of 2n + 1 = N BCn elements. Next consider the case that Rn = Bn . Write the two summations in the bracket of (4.28) as S+ :=

n 

 c sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ),

=1

S− := −

n 

 c sin(γ j u  )e−i(x+u  ) θ(−ei(x+u  ) ; pt ).

(4.30)

=1

If we change the variable of the summations in S+ and S− as − + 1 → , we have S± = ∓

0 

 c−+1 sin(γ j u  )e−i(x±u  )/2 θ(−ei(x±u  ) ; pt ) =: S∓ .

=−n+1

Therefore, 1 {(S+ + S+ ) + (S− + S− )} 2 n 1   c sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ) = 2 =−n+1

S+ + S− =

−

n 1   c sin(γ j u  )e−i(x+u  )/2 θ(−ei(x+u  ) ; pt ), 2 =−n+1

where  c :=  c−+1 for  = −n + 1, . . . , 0. Since u Bn = 2π( − 1/2)/(2n − 1), we see that the term with  = −n + 1 in the first (resp. second) summation is cn sin(γ j π) e−i(x−π)/2 θ(−ei(x−π) ; pt )) − cn sin(γ j π) e−i(x+π)/2 θ(−ei(x+π) ; pt ) (resp. and this is equal to the term with  = n in the second (resp. first) summation. Since cBn /2,  = 1, . . . , n − 1, we can conclude that  cnBn = 1/2 =  S+ + S− = − and hence

n 1  sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ) 2 =−n+2 n 1  sin(γ j u  )e−i(x+u  )/2 θ(−ei(x+u  ) ; pt ), 2 =−n+2

52

4 KMLGV Determinants and Noncolliding Brownian Bridges −γ /2 n 1/8  i p t ( p t ; p t )∞ p t j = sin(γ j u  ) ( pN 2 t ; pN 2 t )∞ N =−n+2   × e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ) − e−i(x+u  )/2 θ(−ei(x+u  ) ; pt ) , 2

L Bj n

(4.31)

where the ranges of the two summations are both {−n + 2, . . . , n} consisting of 2n − 1 = N Bn elements. Now we consider the case that Rn = Cn∨ . Write the two summations in the bracket C∨ of (4.28) as (4.30). Since sin(γu  n ) = sin(γ j π/n) = 0 if  = 0, we can change the ranges of  in S+ and S− in the bracket from {1, . . . , n} to {0, 1, . . . , n} with setting  c0 := 1. We can also rewrite S+ and S− by replacing the variables in the summation as  → −. Then we can show that n 1   c sin(γ j u  )e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ) S+ + S− = 2 =−n

−

n 1   c sin(γ j u  )e−i(x+u  )/2 θ(−ei(x+u  ) ; pt ), 2 =−n C∨

where  c :=  c− for  = −n, . . . , −1. Since u  n = π/n, we see that the term with  = −n in the first (resp. second) summation is − cn sin(γ j π) e−i(x+π)/2 i(x+π) −i(x−π)/2 i(x−π) θ(−e ; pt ) (resp.  cn sin(γ j π) e θ(−e ; pt )) and this is equal to the C∨ C∨ term with  = n in the second (resp. first) summation. Since  cn n = 1/2 =  c n /2,  = 1, . . . , n − 1, we can conclude that −γ /2 n 1/8  i p t ( p t ; p t )∞ p t j = sin(γ j u  ) ( pN 2 t ; pN 2 t )∞ N =−n+1   × e−i(x−u  )/2 θ(−ei(x−u  ) ; pt ) − e−i(x+u  )/2 θ(−ei(x+u  ) ; pt ) , 2

C∨ Ljn

(4.32)

where the ranges of the two summations are both {−n + 1, . . . , n} consisting of ∨ 2n = N Cn elements. Following the calculations similar to those given in (i), we can verify (4.26) from (4.29), (4.31), and (4.32) for Rn = BCn , Bn , and Cn∨ , respectively.  (iii) The proof of (4.26) for Rn = Dn is left for readers as Exercise 4.1. Let det

1≤ j,k≤n

[ Rjkn ( pt )]

 i n  Rn ( pt ), =:  Dn ( pt ),

Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , R n = Dn .

(4.33)

Then ( pt ) ∈ R and ( pt ) = 0 in general by the definition (4.33) with (4.25), and we have the following equalities:

4.3 KMLGV Determinants and Noncolliding Brownian Bridges in [0, π]

53



det [p[0,π] (0, u j ; t, xk )] ⎧ n ⎪ (−i)n e−i =1 x /2 ⎪ ⎪ det [ψ Rj n (ei xk ; pNt )], Rn = Bn , Cn∨ , BCn , ⎪ ⎪ 1≤ j,k≤n ( p ) ⎪ t ⎨ (−i)n Rn i x k = Rn = Bn∨ , Cn , det [ψ j (e ; pNt )], ⎪ 1≤ j,k≤n ( p ) ⎪ t ⎪ ⎪ 1 ⎪ ⎪ det [ψ Dn (ei xk ; pNt )], R n = Dn , ⎩ ( pt ) 1≤ j,k≤n j

1≤ j,k≤n

(4.34)

x ∈ Wn ([0, π]), where u = u Rn given by (4.24), N = N Rn given by (3.9), ( pt ) =  Rn ( pt ) given by (4.33), and the boundary conditions are specified as ⎧ ⎪ ⎨ar,  = aa, ⎪ ⎩ rr,

for Rn = Bn , Cn∨ , BCn , for Rn = Bn∨ , Cn , for Rn = Dn .

(4.35)

Definition 4.2 Fix T > 0. For each Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , Dn , the initial and the final configurations are identical and given by u Rn as (4.24), and the boundary conditions at x = 0 and x = π are specified by (4.35). Then the system of noncolliding Brownian bridges of type Rn is the Markov process (X Rn (t))t∈[0,T ] of n particles on [0, π] with time duration T , such that its multi-time joint probability distribution is determined by the following density function: For an arbitrary m ∈ N and for an arbitrary series of strictly increasing times t0 := 0 < t1 < · · · < tm < T , the multi-time joint probability density of configurations x() ∈ Wn ([0, π]) at times t = t ,  = 1, . . . , m is given by pTRn (t1 , x(1) ; t2 , x(2) ; · · · ; tm , x(m) ) =

 (m)  (−1) () p[0,π] (tm , x j ; T, u k ) , det [p[0,π] (t−1 , x j ; t , xk )]  1≤ j,k≤n p[0,π] (0, u j ; T, u k ) =1

m 

(4.36)

where x(0) := u Rn . In particular, the single-time distribution is described by the probability density pTRn (t, x) = pTRn (0, u; t, x; T, u) :=

det [p[0,π] (0, u j ; t, xk )] det [p[0,π] (t, x j ; T, u k )]

1≤ j,k≤n

det

1≤ j,k≤n

t ∈ [0, T ], x ∈ Wn ([0, π]). We can prove the following.

1≤ j,k≤n  [p[0,π] (0, u j ; T, u k )]

,

(4.37)

54

4 KMLGV Determinants and Noncolliding Brownian Bridges

Proposition 4.3 Let Rn = Bn , Bn∨ , Cn , Cn∨ , BCn , and Dn . For each t ∈ [0, T ], 1 Z Rn 1 = R Z n

pTRn (t, x) =

det [ψ Rj n (ei xk ; pNt )] det [ψ Rj n (ei xk ; pN(T −t) )]

1≤ j,k≤n

1≤ j,k≤n

det [ψ Rj n (ei xk ; pNt )] det [ψ Rj n (ei xk ; pN(T −t) )],

1≤ j,k≤n

(4.38)

1≤ j,k≤n

x ∈ Wn ([0, π]), where N = N Rn given by (3.9) and Z R n = Z R n ( pt , p T ) = π n

n 

h Rn ( pNt , pN(T −t) )

=1

=2

n+n 0

√ π n θ Rn

with n0 =

θ Rn

n 0Rn

⎧ ⎪ ⎨0, = 1, ⎪ ⎩ 2,

( pN 2 T ; pN 2 T )n∞ (4.39) ( pN 2 t ; pN 2 t )n∞ ( pN 2 (T −t) ; pN 2 (T −t) )n∞ Rn = Cn , Cn∨ , BCn , Rn = Bn , Bn∨ , R n = Dn ,

⎧ θ(−1; pN T )θ(−1; pN 2 T ), ⎪ ⎪ ⎪ ⎪ 1/2 ⎪ θ(−1; pN T )θ(−1; pN 2 T )/θ(− pN 2 T ; pN 2 T ), ⎪ ⎪ ⎪ ⎨θ(−1; p )/{θ(−1; p 2 )θ(− p 1/2 ; p 2 )}, NT N T N T N2T = θ Rn ( p T ) = 1/2 ⎪ ; p ), θ(− p NT ⎪ NT ⎪ ⎪ ⎪ ⎪ )/θ(−1; pN 2 T ), θ(−1; p N T ⎪ ⎪ ⎩ 1/2 θ(−1; pN T )θ(−1; pN 2 T )θ(− pN 2 T ; pN 2 T ),

Rn Rn Rn Rn Rn Rn

= Bn , = Bn∨ , = Cn , = Cn∨ , = BCn , = Dn . (4.40)

Proof In (4.34) we replace t by T − t and take the complex conjugate. Notice that by    (2.32), p[0,π] (0, u j ; T − t, xk )] = p[0,π] (t, u j ; T, xk ) = p[0,π] (t, xk ; T, u j ), and that ( pt ) is real-valued by the definition (4.33) with (4.25). We have 

det [p[0,π] (t, x j ; T, u k )] ⎧ n ⎪ i n ei =1 x /2 ⎪ ⎪ det [ψ Rj n (ei xk ; pN(T −t) )], Rn = Bn , Cn∨ , BCn , ⎪ ⎪ 1≤ j,k≤n ( p ) ⎪ T −t ⎪ ⎪ ⎪ ⎪ ⎨ in = det [ψ Rj n (ei xk ; pN(T −t) )], Rn = Bn∨ , Cn , ⎪ 1≤ j,k≤n ( p ) ⎪ T −t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ det [ψ Dn (ei xk ; pN(T −t) )], R n = Dn . ⎩ ( pT −t ) 1≤ j,k≤n j

1≤ j,k≤n

4.4 Noncolliding Brownian Bridges and Macdonald Denominators

55

Then the single-time probability density (4.37) is given by pTRn (t, x) =

1 Z

det [ψ Rj n (ei xk ; pNt )] det [ψ Rj n (ei xk ; pN(T −t) )],

1≤ j,k≤n

1≤ j,k≤n

(4.41)



where Z = ( pt )( pT −t ) det [p[0,π] (0, u j ; T, u k )] ∈ R. By calculating the inte1≤ j,k≤n

grals of both sides of the equality (4.41) with respect to x over Wn ([0, π]), we have the following equalities, where the Andréief identity (4.17) is used: πn 1= Z πn = Z =

Hence Z = π n



d x Rn i x det ψ (e ; pNt )ψ Rn (ei x ; pN(T −t) ) 1≤ j,k≤n [0,π] π   det h Rj n ( pNt , pN(T −t) )δ jk



1≤ j,k≤n

n π n  Rn h ( pNt , pN(T −t) ). Z j=1 j

n 

h Rj n ( pNt , pN(T −t) ). By (3.28) with (3.29) in Proposition 3.4 and

j=1

the fundamental property of the theta function (1.12), (4.39) with (4.40) is verified. Hence the proof of (4.38) is complete. 

4.4 Noncolliding Brownian Bridges and Macdonald Denominators We recall the determinantal identities of Rosengren and Schlosser given by Propositions 3.1 and 3.2 at the beginning of Chap. 3. Then the single-time probability densities of the Brownian bridges given by Propositions 4.2 and 4.3 are written as follows: for t ∈ [0, T ], with e±ix := (e±i x1 , . . . , e±i xn ), A

pT n−1 (t, x) =

( pnt ; pnt )n∞ ( pn(T −t) ; pn(T −t) )n∞ (2π)n ( pn 2 T ; pn 2 T )n∞ n

n

; pnt )θ(r T −t e−i =1 x ; pn(T −t) ) × θ(−|r T |; pnT ) × M An−1 (eix ; pnt )M An−1 (e−ix ; pn(T −t) ), x ∈ Wn ([0, 2π)), (4.42) θ(rt ei

=1

x

56

4 KMLGV Determinants and Noncolliding Brownian Bridges

and ( pNt ; pNt )n∞ ( pN(T −t) ; pN(T −t) )n∞ f Rn √ ( pN 2 T ; pN 2 T )n∞ (2π)n θ Rn × M Rn (eix ; pNt )M Rn (e−ix ; pN(T −t) ), x ∈ Wn ([0, π]),

pTRn (t, x) =

(4.43)

where N = N Rn given by (3.9), θ Rn = θ Rn ( pT ) given by (4.40), and

f Rn

⎧ ⎪ 2, ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨2 ( pNt; pNt )∞ ( pN(T −t) ; pN(T −t) )∞ , Rn = f ( pt , pT ) = 1, ⎪ ⎪ 1/2 1/2 ⎪ ( pNt ; pNt )∞ ( pN(T −t) ; pN(T −t) )∞ , ⎪ ⎪ ⎪ ⎩ 4,

Rn = Bn , Rn = Bn∨ , Rn = Cn , BCn , Rn = Cn∨ , r

(4.44)

R n = Dn .

Here we take the following limit: t → ∞ and T − t → ∞,

(4.45)

Then pt → 0, pT −t → 0, rt → 0, r T −t → 0, and we have the following limit distributions, which are independent of time t: 1 |W An−1 (eix )|2 t→∞, (2π)n T −t→∞    xk − x j 1 , x ∈ [0, 2π)n , = A sin2  2 Z n−1 1≤ j