17 July 2017

## Percolation

From wikipedia: In physics, chemistry and materials science, percolation refers to the movement and filtering of fluids through porous materials. Original problem was described by Broadbent and Hammersley in 1957: Will the water reach the center of the porous rock when it is submerged under water?

Percolation theory studies such models. It can be modelled in two ways.

#### Bond percolation

An edge or a bond between two sites may be open with probability `p`

or closed with
probability `1 - p`

. What is the probability that a path over open bonds exists from one side to another?

#### Site percolation

Another way to model is to say that a site itself is open with probability `p`

or closed
with probability `1 - p`

. What is the probability that
a path over open sites exits from one side to another?

*Note: in this post connectivity is examied between left and right sides.*

Example of site percolation simulation with `p = 0.59`

(blue site are closed, orange are open):

As you can see, there is no path from left to right over open (orange) sites. But if one site in lower right becomes open

then site percolation is observed:

#### Critical percolation threshold

In case of infinite network, for any given probability `p`

the probability that an infinite open cluster exists
is either zero or one meaning that there must be critical `p`

below which the probability
of an infinite open cluster is zero and above is one._{c}

Example of site percolation probability given probability `p`

to enable a single site for n = 400:

As you can see the critical probability `p`

for 2D site percolation is between _{c}`0.59`

and `0.60`

.

You can play with site percolation here.