Current Developments in Mathematics
Introduction to KPZ
This is an introductory survey of the Kardar-Parisi-Zhang equation (KPZ). The first chapter provides a non-rigorous background to the equation and to some of the many models which are supposed to lie in its universality class, as well as the predicted, non-standard fluctuations. The second chapter provides a rigorous introduction to the stochastic heat equation, whose logarithm is the solution of KPZ, as well as some of the known methods for proving convergence of discrete growth models and directed polymer free energies. Finally, we end with a sketch of the derivation of exact formulas for the one-point distributions of KPZ at finite time for special initial data, from the Tracy-Widom formulas for asymmetric exclusion.
2010 Mathematics Subject Classification
60B20, 60F05, 60H15, 60K35, 82C22, 82C44