Stochastic Geometry and Field Theory: From Growth Phenomena to Disordered Systems

Coordinators: David Alexander Ellwood, Ilya Gruzberg, Pierre Le Doussal, Andreas W.W. Ludwig, Scott Sheffield, Paul Wiegmann, Kay Joerg Wiese

Many important physical phenomena reveal stochastic or random geometrical objects and shapes. Among them are fluctuating domain boundaries in statistical mechanics, growing patterns in non-equilibrium processes, and fluctuating surfaces studied in string theory. Random geometrical objects naturally arise in the physics of disordered systems and random media. In many interesting cases they are fractal in nature. Their statistics may be studied by methods of conformal field theory and the renormalization group.

The workshop focuses on a wide class of physical problems involving stochastic or random geometry and novel analytical techniques developed to attack these problems.

Perhaps the most spectacular recent development is the discovery of the Stochastic Loewner Evolution (SLE) and the ensuing revitalization of the study of 2D critical phenomena as a stochastic evolution of geometry. This remarkable development links conformal field theory to probability theory and complex analysis. Kinetic growth phenomena like Diffusion-limited aggregation and Laplacian growth are all close cousins of SLE and have been studied in two dimensions using conformal maps, their evolution and iterations.

Conformal Field Theory and methods inspired by conformal invariance proved to be productive in problems of disordered electronic systems and localization, including quantum-Hall plateau transitions, and multifractal properties of wave functions.

Functional Renormalization methods for disordered systems allow to treat quantitatively problems which are out of reach for more conventional field theories. Such a description may contain shocks and singularities within a field theoretical framework, and account for glassy behavior. This includes the Burgers and KPZ equations, for which exact results from combinatorics and random matrix theory became available recently.

The goal of the workshop is to bring together physicists and mathematicians from different disciplines, in order to communicate the new ideas and methods and to promote cross-fertilization between different areas of mathematics and physics. The theme of the workshop is not so much a specific set of physical phenomena. With this idea in mind, we encourage people from a wide variety of disciplines to apply, who are interested in such a more methods-oriented workshop.