They used a cellular automaton model to demonstrate this process, using three basic parameters: the probability of civilization forming, its likely life span, and the extended life span it would enjoy should it come into contact with another civilization.
And if the values for those parameters are just right - "finely tuned" in physics speak - then a kind of phase change occurs. But instead of going from a solid to a liquid (or a gas), the universe goes from one in which civilizations scattered across the cosmos are unlikely to meet, to one in which they can. Who knows, perhaps even a federation of various civilizations could evolve - assuming there isn't an interstellar war, with one civilization wiping out another and taking over their planet (the stuff of good science fiction for decades).
This might explain Fermi's Paradox: we don't have alien visitors (or communications) because our universe hasn't undergone that critical phase change - i.e., we don't live in that finely tuned universe where the parameters are just right to give rise to mixing civilizations. Or we haven't been around long enough yet for the mixing to occur.