This models an evolving set in the plane that is dual to the
random walk
Prob(Xn+1=y | Xn=x) =
{
p if x↔y
γ if x=y
0 otherwise.
To make entering numbers a little easier, instead of p and γ,
you just enter r=γ/p, the rate at which the walk loops back
on itself.
| Self Loop Rate: | 0≤r | |
|---|---|---|
| Percolation Probability: | 0<p≤1 | |
| Couple a second evolution: | ||
| Coupled percolation probability: | 0<p≤1 | |
| Underlying Graph: | ||
| Evolve on a cylinder: | ||
| Steps per second: | ||
| Steps taken: | 0 |
When p = 1 for an uncoupled percolation not on a cylinder,
the result for the first three lattices is generally a
semi-regular 2k-gon.
In this case, it is faster to just follow
a corner of the evolving shape,
which converges to a
stochastic diffusion.
The only case where this shape doesn't happen is
when 0<r<1 on the hexagonal lattice.
In this case, when 0<r≪1, the
result is interesting.
| Neither Not the cluster of the origin | The larger cluster of the origin | Both clusters of the origin | |||
|---|---|---|---|---|---|
| Neither Not the percolation cluster | Neither Not the evolved shape | ||||
| Larger The percolation cluster | Neither Not the evolved shape | Neither Not the evolved shape | Larger The evolved shape | ||
| Both percolation clusters | Neither evolved shape | Neither evolved shape | Larger evolved shape | Neither evolved shape | Larger evolved shape |
| Smaller evolved shape | Both evolved shapes | ||||
"Larger" and "smaller" refer to the
larger and smaller percolation probabilities being used, not
(necessarily) to which shape is bigger.
The evolving set is initially
the entire left side, while
the top wraps around to the bottom.
a single point at the origin.
Critical percolation occurs at p
≈ 0.5927.
≈ 0.6962.
= 1/2.
≈ ?