Extremal point of infinite clusters in stationary percolation

Abstract

It is well known that in stationary percolation an innite component cannot have a nite number of extremal points in a certain direction In this note we investigate whether or not an innite cluster can have innitely many extremal points in a certain direction To make this question at all interesting it is necessary and natural to simultaneously ask for an innite path in the opposite direction It turns out that the answer depend on the dimension of the model and on the question whether or not the model has socalled nite range