Geometric Aspects of Random Spatial Systems

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

0072217 Pisztora The investigator proposes a number of inter-related projects to study spatial random systems which are of interest both in probability theory and statistical physics. The projects are concerned with regular, invasion and FK percolation, Ising-Potts models and network flows. In percolation and invasion percolation certain structural properties of large clusters will be studied at or near the critical point in two or higher dimensions. In a further project the relation between characteristic length and renormalization will be investigated. A project focuses on phase coexistence in Ising-Potts models and it is concerned with the microscopic properties of the Wulff droplet and of other interfaces, and with the analysis of phase coexistence in the presence of several phases. Decay of correlations and the relaxation of boundary effects in FK percolation (and indirectly in Potts models) is the target of two further projects. The study of capacitated networks is proposed by using sophisticated large deviation methods which have proven to be decisive in other context. The aim of this project is to gain insight and achieve understanding of certain fundamental systems having their origin in statistical and condensed matter physics, materials science, chemistry and earth sciences. Success in solving some of the proposed problems would lead to some of the following benefits: 1) the derivation of rigorous knowledge about fundamental physical systems of theoretical and practical relevance, 2) a deeper insight into the mechanisms responsible for the behavior of such systems, 3) the development of new mathematical methods and tools which, apart from yielding answers to questions raised in other fields, might benefit and enrich probability theory itself.