Invariant Measures for Random Growth Processes

University Of Washington, Seattle WA

Investigators

Abstract

In this proposal we will study first-passage percolation, last-passage percolation and Richardson's growth model. In contrast to the typical manner of studying these as subadditive processes we will view them as interacting particle systems. We will consider several ways of turning first-passage percolation into a Markov process. These processes are new examples of interacting particle systems. We will study these processes on many graphs including Z^d and Delaunay triangulations in R^d. For each of these systems we propose to classify the invariant measures. We feel that studying first passage percolation in this manner is not only this is interesting in its own right, it will also lead to the resolution of conjectures about the original first-passage percolation and Richardson's growth model. These include showing that the number of infinite geodesics in first-passage percolation is infinite and that the boundary of the limiting shape has no sharp corners. For the two type Richardson's model we feel this method will show that coexistence is impossible when the two infections have different speeds. We will also study the connection between last-passage percolation on Z^3 and invariant measures for certain dynamics (called "totally asymmetric hexagon flipping") on lozenge tilings of the plane. We will analyze the invariant measures for the totally asymmetric hexagon flipping process and use the invariant measures to study the limiting shape of last-passage percolation in Z^3. In this proposal we study several processes which model the growth of different objects in space and time. These objects may be different species or populations. In the models we look at the different objects are expanding and competing for territory. Our models also assume that two different objects cannot occupy the same area at the same time. We will study under what rules of growth it is possible that more than one of the objects survive for an arbitrary large time and conversely under which rules of growth does one object conquer all of the others. In this proposal will seek to analyze the processes in the framework of interacting particle systems which was developed in the 1980s and 1990s.