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Percolation (probability)
The process of a liquid infiltrating sand, or a blight spreading through the trees in a forest,
can be modeled by a mathematical approach known as percolation. Percolation takes place
on a graph.
To model percolation on a graph , we choose some probability p. Each edge will be
“open” with probability p, and “closed” with probability 1 p. Once the open and closed
edges have been determined, we can start at any vertex v and form the “cluster” which
consists of all vertices connected to v by using only open edges. The basic question is: what
is the expected size of ? With what probability will it be finite (or infinite)? More generally,
what is the “mass spectrum”, that is, the probability distribution of sizes of clusters? How
do all these properties vary with p?
A good place to start is with the “regular tree of valence d”: the unique graph such that
every vertex has exactly d edges ending there, which join it to other vertices, and such that
there are no closed paths. For d = 2, one can label the vertices by integers k 2 Z, and then
k is joined to k ± 1 by edges.
There are of course other graphs one can consider, which are more realistic for applications, such as the square lattice: vertices are Z2 , each vertex (x, y) is connected to (x ± 1, y)
as well as to (x, y ± 1).
There is another process similar to percolation, called invasion. In this process, we
assign to each edge a number ce 2 [0, 1], chosen at random. We form an infinite sequence
of subgraphs 1 , 2 , . . . as follows. 1 contains only a single vertex {v}, which is our fixed
starting point. Then, at step k, look at all the edges which do not lie in k , but which are
adjacent to it. Among these edges take the one with the lowest probability, which we denote
by ck , and add that edge (with its other endpoint) to our graph, thereby forming k+1 . Look
at the sequence c1 , c2 , . . . : what is its “lim sup”? Is there any relation to the behaviour of
percolation?