Elliptic Extensions in Statistical and Stochastic Systems

Description

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