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

ISBN-13: 9789811002748 / Angielski / Miękka / 2016 / 141 str.

Makoto Katori

Opis

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 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 Tracy Widom distribution is derived. "

Spis treści

1 Bessel Process1.1 One-Dimensional BrownianMotion (BM) 1.2 Martingale Polynomials of BM 1.3 Drift Transform 1.4 Quadratic Variation 1.5 Stochastic Integration 1.6 Ito's Formula 1.7 Complex Brownian Motion and Conformal Invariance 1.8 Stochastic Differential Equation for Bessel Process 1.9 Kolmogorov Equation 1.10 BES(3) and Absorbing BM1.11 BES(1) and Reflecting BM 1.12 Critical Dimension Dc = 2 1.13 Bessel Flow and Another Critical Dimension Dc = 3/2 1.14 Hypergeometric Functions Representing Bessel Flow Exercises References 
2 Schramm--Loewner Evolution (SLE) 2.1 Complexification of Bessel Flow2.2 Schwarz--Christoffel Formula and Loewner Chain 2.3 Three Phases of SLE2.4 Cardy's Formula 2.5 SLE and Statistical Mechanics Models Exercises References 
3 Dyson Model 3.1 Multivariate Extension of Bessel Process 3.2 DysonModel as Eigenvalue Process 3.3 Dyson Model as Noncolliding Brownian Motion 3.4 Determinantal Martingale Representation (DMR) 3.5 Reducibility of DMR and Correlation Functions 3.5.1 Density Function rx(t, x) 3.5.2 Two-Time Correlation Function rx (s, x;t, y) 3.6 Determinantal Process 3.7 Constant-Drift Transform of Dyson Model 3.8 Generalization for Initial Configuration with Multiple Points 3.9 Wigner's Semicircle Law and Scaling Limits 3.9.1 Wigner's Semicircle Law3.9.2 Bulk Scaling Limit and Homogeneous Infinite System <3.9.3 Soft-Edge Scaling Limit and Spatially Inhomogeneous Infinite System3.10 Entire Functions and Infinite Particle Systems 3.10.1 Nonequilibrium Sine Process 3.10.2 Nonequilibrium Airy Process 3.11 Tracy--Widom Distribution 3.11.1 Distribution Function of Maximum Position of Particles 3.11.2 Algebraic Calculation 3.11.3 System of Differential Equations 3.11.4 Nonlinear Third-Order Differential Equation 3.11.5 Soft-Edge Scaling Limit 3.11.6 Painleve II and Limit Theorem of Tracy and Widom 3.12 Beyond Determinantal Processes Exercises 
References Solutions Index