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

By Makoto Katori

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 References Solutions Index

• Format: Paperback
• Language: English
• ISBN: 9811002746
• ISBN13: 9789811002748
• Release Date: February 2016
• Publisher: Springer
• Length: 141 Pages
• Weight: 0.49 lbs.
• Dimensions: 0.3" x 6.1" x 9.2"