Growth of random surfaces

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The proposal focuses on a class of random growth models of one and two-dimensional interfaces. The growth is local, it has a smoothing mechanism, and the speed of growth may depend on the local slope of the interface.The central role is played by models that enjoy a rich algebraic structure.In many cases this is reflected through determinantal formulas for the correlation functions. Large time asymptotic analysis of such formulas is expected to reveal asymptotic features of the emerging interface in different scales. The main goal of the project is to produce and collect various pieces of information to form a unified picture of the asymptotic behavior of large randomly growing interfaces.The proposal focuses on the study of large time behavior for various models of random interface growth on the plane and in the three-dimensional space. Such models appear naturally in both mathematics and physics, for example when one considers ideal crystal melting, or molecular condensation on a substrate, or traffic evolution. Accurate analysis at large times is typically very difficult, and we concentrate on models with additional algebraic structure that originates in seemingly unrelated parts of mathematics. This structure helps to discover new phenomena that tend to be universal, i.e. present in a much wider range of models. As a result, using sophisticated mathematical tools, we find new universal laws that play the role of the famous bell-shaped curve, and that can often be observed later through physical and numerical experiments.