Long range percolation in stationary point processes

We consider a continuum percolation model in ?d, d ? 1 in which any two points of a stationary point process are connected with a probability which decays exponentially in the distance between the points. We give sufficient conditions for the (non)-existence of a phase transition. We also give examples of processes which show that it is impossible to write down a theorem which relates the critical parameter value of a process to its density. Finally, we show that uniqueness of the infinite cluster is still valid in this general setting. © 1993 John Wiley & Sons, Inc.