Deterministic Percolation

This paper examines percolation questions in a deterministic setting. In particular, I consider , the set of elements of Z ² with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result. Received: 1 November 1998 / Accepted: 1 April 1999