The KardarParisiZhang equation and universality class
Abstract
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the KardarParisiZhang (KPZ) universality class and underlying it is, again, a continuum object  a nonlinear stochastic partial differential equation  known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact onepoint distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 DOI:
 10.48550/arXiv.1106.1596
 arXiv:
 arXiv:1106.1596
 Bibcode:
 2011arXiv1106.1596C
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 57 pages, survey article, 7 figures, addition physics ref. added and typo's corrected