The Kardar-Parisi-Zhang equation and universality class

October 24 to October 28, 2011

at the

American Institute of Mathematics, San Jose, California

organized by

Ivan Corwin and Jeremy Quastel

Original Announcement

This workshop will be devoted to the study of the the Kardar-Parisi-Zhang equation and universality class.

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 continuum properties (such as distribution functions) and expanding the breadth of its universality class. Recently, a new universality class has emerged to describe a host of important physical and probabilistic models (such as one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display unusual scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- now a non-linear stochastic partial differential equation -- which is known as the KPZ equation.

The purpose of this workshop is to build on recent successes in understanding the KPZ equation and its universality class. There are two main focuses:

  1. Studying the integrability properties and statistics of the KPZ equation. Surprisingly, it is possible to give exact formulas for certain statistics associated with this non-linear stochastic PDE -- such as its one-point distribution. We seek to understand the extend to which one can perform exact calculations using the rigorous Bethe Ansatz approach of Tracy-Widom; the non-rigorous replica methods; or the tropical Robinson-Schensted-Knuth correspondence. We also seek to elucidate the poorly understood connection with random matrices.

  2. Extending the universality of the KPZ equation. Though it recognized physically as universal, the KPZ equation has only been shown to rigorously occur as the scaling limit of a few special models, for example, weakly asymmetric simple exclusion processes, and weakly rescaled polymers. In large part this is due to the fact that the KPZ equation is not well-posed, and is presently only defined through the Hopf-Cole transform of the well-posed multiplicative stochastic heat equation. A good well-posedness theory should provide a route to proving scaling limits to the KPZ equation for a wider class of models.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Macdonald processes
by  Alexei Borodin and Ivan Corwin
Stationary correlations for the 1D KPZ equation
by  Takashi Imamura and Tomohiro Sasamoto,  J. Stat. Phys. 150 (2013), no. 5, 908-939  MR3028391
From duality to determinants for q-TASEP and ASEP
by  Alexei Borodin, Ivan Corwin, and Tomohiro Sasamoto,  Ann. Probab. 42 (2014), no. 6, 2314-2382  MR3265169
Free energy fluctuations for directed polymers in random media in 1+1 dimension
by  Alexei Borodin, Ivan Corwin, and Patrik Ferrari,  Comm. Pure Appl. Math. 67 (2014), no. 7, 1129-1214  MR3207195