Introduction to KPZ. (English) Zbl 1316.60019
Jerison, David (ed.) et al., Current developments in mathematics, 2011. Papers based on selected lectures given at the current development mathematics conference, Harvard University, Cambridge, MA, USA, November 2011. Somerville, MA: International Press (ISBN 978-1-57146-239-8/pbk). 125-194 (2012).
Summary: 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.
For the entire collection see [Zbl 1256.00015].
For the entire collection see [Zbl 1256.00015].
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |
| 82-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics |
| 60F05 | Central limit and other weak theorems |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 82C44 | Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics |
