9704901 Pisztora The investigator proposes to study several inter-related projects in random spatial models, more specifically in ordinary and invasion percolation, in Ising and random cluster models, and in first passage percolation. In ordinary and invasion percolation the critical behavior of the chemical distance will be investigated. Another project is concerned with the large deviation behavior of the random resistor network. In Ising and random cluster models the investigator intends to study the rigidity (roughness) of certain phase boundaries. In particular, questions related to roughening in higher dimensional percolation models will be studied. The investigator also intends to work on questions concerning the shape of certain interfaces. The project in first passage type models includes work on comparison principles between minimizing paths and certain random walks, and (jointly with another colleague) on the confinement property of directed polymers in a random environment. The aim of this project is to study certain geometric aspects of random spatial systems. In some random systems geometric objects (in this case surfaces) which are deterministic on a macroscopic scale arise naturally. Examples of such objects are phase boundaries and droplets ("crystals") in spin systems or the frontier of the invaded region in first passage percolation, which is a basic model for randomly evolving (growing) surfaces. Both the (asymptotic) shapes and the fluctuations of these objects are of prime concern with great theoretical and practical interest. Many of the proposed problems and models have their origin (and are still studied) in statistical physics, materials science, chemistry, and condensed matter physics. The project's final goal is to gain some insight into the behavior of random surfaces - objects which seem to appear just about everywhere in nature, but are known to be difficult to understand from the theoretical (mathematical) p oint of view.
