*Percolation* is a fascinating stochastic process which can model many physical effects, as well as being interesting in its own right. Like many such random models, it displays a variety of "threshold effects": as a controlling parameter changes continuously, different qualitative effects occur.

I exhibit the simplest model of percolation. This model is closely related to the *Ising model* of statistical mechanics, which describes how spins align themselves to create magnetism. But percolation can also be used in many other scenarios: to describe global connectivity properties despite link failure in a communications network, to describe social interaction, or whatever else strikes your fancy. In fact, the coffee-related name "percolation" is apparently the result of the initial effect which it was used to model: how gas in a coal mine permeates the filter of a gas mask. In all these physical effects, a one parameter changes value gradually, leading to a sudden "collapse" of the system into a different phase. This is a *phase transition*, and we can see the mathematics of one such phase transition in some detail.

I deal mostly with percolation on infinite graphs. The study of percolation on finite graphs has a distinct flavour, tends to be more related to Computer Science and applications, and, while sharing many properties, is even harder to analyze.

## Definition

Let G=(V,E) be a graph. Percolation on G (the name "bond percolation" is often used instead, to indicate that edges are erased but vertices kept) is a set of random variables X_{e} belonging to each edge e∈E, each X_{e} taking on the values {0,1}. When X_{e}=1, we say that edge e is "open"; when X_{e}=0, e is said to be "closed".

We shall take all X_{e}'s to be IID (which see).

In other words, fix some probability 0≤p≤1. For every edge in E, flip a coin which comes up heads with probability p; delete ("close") the edge if the coin came up tails. The resulting graph is (a sample of) percolation on G, with edge probability p.

Now suppose G is infinite and a transitive graph (which see). Consider the event C={there exists an infinite connected open cluster} (i.e. an infinite set of vertices A⊆V, such that there is a path of open edges between any two elements of A). Note that C is a tail event of the variables X_{e}, so by Kolmogorov's 0-1 Law we have either that P(C)=0 or P(C)=1. Which of the two cases depends on the choice of the graph G and of the parameter p.

Critical percolation is the study of the phase transition between C occurring and not occurring.

To give a taste of the theory, here are some other events we may consider:

- For some v∈V, C
_{v}={there exists an infinite connected open cluster containing v}. When G is transitive, P(C_{v})=P(C_{u}) for any 2 vertices u,v, and it turns out that P(C_{v})>0 iff P(C)=1.
- Define θ(p) to be P(C) when we perform percolation with edge probability p. What is the behaviour of θ(p)? Is it continuous?
- Assuming P(C)=1 --
*how many* infinite open clusters are there?
- Assuming P(C)=0 -- what is the expected size of an open cluster?

Many of these numbers are

*unknown*. Even more frustratingly, many of the

answers are "known" (from

Physics), but we are unable to prove them!