Extremal points of infinite clusters in stationary percolation

It is well known that in stationary percolation, an infinite component cannot have a finite number of extremal points in a certain direction. In this note, we investigate whether or not an infinite cluster can have infinitely many extremal points in a certain direction. To make this question all interesting, it is necessary (and natural) to simultaneously ask for an infinite path in the opposite direction. It turns out that the answer depends on the dimension of the model, and on the question whether or not the model has the so-called finite range. (C) 1999 Elsevier Science B.V. All rights reserved.