Percolation was introduced by Broadbent and Hammersley in 1957. The
simplest version to describe is on the Euclidean
lattice **Z**^{d}. Let *p* be a fixed probability between 0
and 1. Each bond in **Z**^{d} is retained with
probability *p*, and removed with probability *1-p*,
independently of all the others. The percolation cluster containing a
point *x*, denoted *C(x)*, consists of those points which can be
reached from *x* by a path of retained bonds. There is a critical
value *p*_{c}∈(0,1) such that if *p < p*_{c}
then all clusters are finite, while for *p > p*_{c} there is
an infinite cluster.

Random walks on percolation clusters were introduced by De Gennes in 1976:
he called this the problem of 'the ant in the labyrinth'. If *p =
p(n,x,y)* is the probability that a random walker ('the ant'), starting
at *x*, is at *y* at time *n*, then *p* describes
diffusion of heat on the cluster.

For the supercritical phase (*p > p*_{c}) this problem is now
quite well understood, and *p(n,x,y)* converges to a Gaussian
distribution as *n→∞*. PDE techniques introduced by
Nash in the 1950s, play an important role in some of the arguments.

The critical case *p = p*_{c} is much harder, since the
clusters have fractal properties. One expects that *p(n,x,x) ∼
x*^{- ds/2}, where *d*_{s} is called the spectral
dimension of the cluster. Alexander and Orbach conjectured in 1982
that *d*_{s} = 4/3 in all dimensions: this has recently been
proved in some high dimensional cases.