Random gradient models with degenerate potential (12rit185)

Arriving in Banff, Alberta Sunday, September 9 and departing Sunday September 16, 2012


(University College London)

Feng Yu (University of Bristol, Department of Mathematics, UK)


The Banff International Research Station will host the "Random gradient models with degenerate potential" workshop from September 9th to September 16th, 2012.

Gradient models are a class of models arising in the study of random interfaces and of elasticity theory. A lot is known about such
models when the interaction satisfies certain strict convexity assumptions and when no disorder is present in the system. For
models which are more realistic approximations of the physical phenomena involved, such as models with non-convex interactions
or models with disorder, much less is known and there are still many interesting open problems.

The main goal of our project is to study interface models with degenerate disorder, which are more realistic approximations of
the physical phenomena involved as they take account of impurities in the system, which can affect, for example, the potentials,
or the configurations of the system on which the interface appears. In the model we propose to study the interaction is random
and may be $0$ with a fixed probability. Our model effectively becomes a gradient interface model defined on a percolation
configuration, and has exciting connections to random walks with random conductances. We plan to investigate
questions of existence of infinite volume Gibbs measures, uniqueness, central limit theorem, and decay of correlations
under various assumptions on the potentials. Degeneracy poses serious challenge in this case, although that also makes the problem extremely interesting.
We expect that by studying the degenerate problem we will develop new techniques, useful for other gradient models of interest,
and obtain new valuable insight into the behaviour of the system without disorder.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).