Each square is a star system. Colonists from the alien homeworld settle every empty system next to it, to the north, south, east and west. Each new colony then has a probability P of becoming a colonizing civilization (bright) that sends out colonists of its own, and otherwise becomes a non-colonizing civilization (pale) that stays home. The green Solar System is the target. Set P, press Start, and see whether the aliens reach us. Click the grid to move the Solar System or the homeworlds. A single run is mostly luck, so use the trials below to get the odds.
By Claude Opus
The standard version of the Fermi Paradox goes like this. If even a small fraction of the galaxy's stars produced civilizations that travel between stars, one of them should have colonized the whole galaxy in a few million years, which is a blink compared to the galaxy's age. So where is everybody? Most answers assume that some step fails: life is rare, intelligence is rare, or civilizations destroy themselves. Geoffrey Landis proposed an answer in which plenty of civilizations colonize, and we still never see them. He presented it at a NASA symposium in 1993 and published it in the Journal of the British Interplanetary Society in 1998.
His model rests on two assumptions. First, there is a maximum distance over which colonizing another star is practical, so colonies only settle their near neighbors. Second, because messages between stars take years or centuries, each colony quickly develops its own culture, independent of whoever settled it. Some colonies will want to expand and some won't. Landis gave each colony a probability P of becoming a colonizer. Colonizers settle all of their unoccupied neighbors. Non-colonizers stay put, and they block the way: a cluster of colonies can only grow through its colonizers, so it ends up surrounded by a wall of civilizations that stopped expanding.
That turns galactic colonization into a percolation problem. Percolation theory was developed in the 1950s to describe things like water seeping through porous rock, and its central result is a sharp threshold. Below a critical probability, every cluster is finite and colonization always dies out, often after a handful of systems. Above it, colonization can spread without limit, but it leaves empty voids behind, sealed off by non-colonizers. Right at the critical value, both the colonized regions and the empty ones can be of any size at all. If the real galaxy sits anywhere near that value, there could be huge regions no one ever settles, and we could be in one of them.
The grid here is flat, and each system has four neighbors. That is the simplest arrangement, and it is why the clusters grow in blocky, diamond-ish shapes rather than circles. On this grid the critical value of P is about 0.593. Landis worked on a three-dimensional cubic grid, where each system has six neighbors and the critical value is about 0.31. The number depends on how many neighbors are in reach, but the behavior on either side of it is the same.
The green Solar System, the multiple homeworlds and the trials are additions, not part of Landis's paper. The Solar System is already occupied, by us, so the aliens can't settle it; the question is only whether one of their colonizers ends up next door. When two alien civilizations meet, neither can settle a system the other already holds, so they stop at a border. A generation here is one hop between stars. The model says nothing about how long a hop takes, which could be centuries or hundreds of thousands of years.
To see the threshold for yourself, run trials at P = 0.55, 0.59 and 0.63 without moving anything. The chance of reaching the Solar System climbs from almost nothing to almost certain over a few hundredths of P. Later papers have extended the model, allowing colonization along diagonals and using the actual positions of the stars within about 130 light years of the Sun, and they come to similar conclusions: whether anyone reaches us depends heavily on parameters we don't know.