Abstract
If even a very small fraction of the hundred billion stars in the galaxy are home to technological civilizations which colonize over interstellar distances, the entire galaxy could be completely colonized in a few million years. The absence of such extraterrestrial civilizations visiting Earth is the Fermi paradox.
A model for interstellar colonization is proposed using the assumption that there is a maximum distance over which direct interstellar colonization is feasable. Due to the time lag involved in interstellar communications, it is assumed that an interstellar colony will rapidly develop a culture independent of the civilization that originally settled it.
Any given colony will have a probability P of developing a colonizing civilization, and a probability (1-P) that it will develop a non-colonizing civilization. These assumptions lead to the colonization of the galaxy occuring as a percolation problem. In a percolation problem, there will be a critical value of the percolation probability, Pc. For P<Pc, colonization will always terminate after a finite number of colonies. Growth will occur in "clusters," with the outside of each cluster consisting of non-colonizing civilizations. For P>Pc, small uncolonized voids will exist, bounded by non-colonizing civilizations. When P is on the order of Pc, arbitrarily large filled regions exist, and also arbitrarily large empty regions.
The Fermi Paradox: An Approach Based on Percolation Theory
NASA Lewis Research Center,