Whatever Happened to Percolation Theory?

Many years ago I was quite interested in mathematical percolation theory. But over the past 10 years or so I haven’t seen many new research papers published on the topic.

This figure illustrates some of the main ideas of percolation:

If some edges are randomly connected, how likely is it that there’s a path from the top to the bottom? In this example, there’s a 7×7 square grid. This creates 84 up-down and left-right edges. Edges were connected randomly with p = 0.4 (which means that the probability that an edge wasn’t connected is 0.6). After the random connections were added, there were 27 out of 84 edges connected, so the actual probability of a connection in this particular grid is 27/84 = 0.32.

The resulting grid does in fact have a connection from top to bottom. The connected path contains 12 of the 27 connected edges so 12/27 = 0.4 of the edges are part of the connection.

There has been a huge amount of research done on percolation theory. One of the results is, for a square grid, that if the probability of connection is 0.5, then there is likely a connected path from top to bottom. Therefore, in this example, it was a bit lucky that a connected path exists.

Note: Let me point out that I have (deliberately) butchered several of the vocabulary terms and a few of the ideas to keep the explanation simple and understandable. If you are a mathematician who works with percolation theory, please don’t slaughter me in the Comments to this post.

As with any math topic, there are gazillions of generalizations. There are many kinds of grids (more accurately lattices), many kinds of connections (bond vs. site, etc.), 3D versions, n-dimensional versions, and on and on. Percolation theory has many relationships with graph theory, fractals, lattice theory, cellular automata, and other areas of math.

So, back to my original question: Why are there so few new research papers on percolation theory? First, maybe there are lots of new research papers on percolation and I’m just not seeing them. But if the number of research papers is in fact declining, I suspect that either 1.) percolation theory has not proven to be particularly useful in applied areas (such as oil moving through porous rock or materials science), or 2.) all the main ideas in percolation theory have been explored. But this is speculation on my part.

Percolation means, “the slow passage of a liquid through a filtering medium”. But in informal usage percolation means giving some thought to a plan or idea. Here are four examples from an Internet search for “bad idea crime” where criminals should have percolated a bit more. (Interestingly, none of stories with these photos identified the criminal — perhaps too embarrassing). Left: A man in a wheelchair in Brazil tries to rob a store while holding a gun with his feet. Did not go well. Left-Center: A man in Philadelphia tries to rob a store using a banana. Not a good idea. Right-Center: A man in Missouri robs a convenience store but a few hours later he tried to grab a gun away from a police officer. Did not go well. Right: A woman in Ohio tried to diguise herself using a cow costume. The mugshot shows her plan needed more thought.

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