Introduction to Kpz

1. A physical introduction 2 1.1. KPZ/Stochastic Burgers/Scaling exponent 2 1.2. Physical derivation 3 1.3. Scaling 3 1.4. Formal invariance of Brownian motion 4 1.5. Dynamic scaling exponent 6 1.6. Renormalization of the nonlinear term 6 1.7. Cutoff KPZ models 7 1.8. Hopf-Cole solutions 8 1.9. Directed polymers in a random environment 11 1.10. Fluctuation breakthroughs of 1999 12 1.11. The Airy processes 15 1.12. Predicted fluctuations for the KPZ universality class 18 1.13. Tracy-Widom formulas for ASEP 19 1.14. Exact results for KPZ 20 1.15. KPZ universality, or universality of KPZ? 21 1.16. KPZ fixed point 22 2. A mathematical introduction 24 2.1. White noise and stochastic integration in 1 + 1 dimensions 24 2.2. Wiener chaos 27 2.3. The stochastic heat equation 27 2.4. Mild solutions 28 2.5. Martingale problem 30 2.6. Chaos representation 31 2.7. Regularity 32 2.8. Comparison 33 2.9. Positivity 33 2.10. The continuum random polymer 36 2.11. Universality of KPZ and the continuum random polymer: The weakly asymmetric limit 37 2.12. Intermediate disorder regime for directed random polymers 37 2.13. Tightness of the approximating partition functions 39 2.14. Asymmetric exclusion 41 2.15. Weakly asymmetric limit of simple exclusion 43 2.16. Steepest descent analysis of the Tracy-Widom step Bernoulli formula 48 References 52