A percolation cluster, especially near p_{c}
is an object of interest independently of the percolation question.
DeGennes termed the problem where some particle diffuses on a cluster
as
``the ant in the labyrinth''. A diffusing particle selects one of its
neighbours randomly and goes there. We call the average distance traveled by
the ants R(t) .

For p well above p_{c}
this is normal diffusion with R ^{2} proportional to t .

For p well below the critical point the diffusion
approaches a constant value.

Near the threshold there is anomalous diffusion,
which was of intense interest in the 80's. Time permitting I will return to this
next week.

A variant is to set the (until now) absent bonds to be normally conducting and the (until now) connected ones superconducting. The walkers here are called ``termites'' and a whole new set of phenomena can be studied.

The ant concept has two applications of immediate interest to us:

one is an easy way to introduce directed percolation - see the next page

the second is that it was via ant diffusion simulations that Amnon Aharony and I stumbled over a set of models that we called diffusion percolation that
turned out to be analogous to bootstrap
percolation which has remained an area of interest both in
applications and theory
until today. (Bootstrap percolation was independently
discovered by several groups, including Ilan Reiss at the Technion.)
Some bootstrap applications even relate to Condensed Matter Physics although others are in Computer Science.