The Kardar-Parisi-Zhang equation and universality class

October 24 to October 28, 2011

at the

American Institute of Mathematics, Palo Alto, California

organized by

Ivan Corwin and Jeremy Quastel

This workshop, sponsored by AIM and the NSF, 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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