# ssa2d: Site cluster on Square Anisotropic 2D lattice with... In SPSL: Site Percolation on Square Lattices (SPSL)

## Description

`ssa2d()` function provides sites labeling of the anisotropic cluster on 2D square lattice with Moore (1,d)-neighborhood.

## Usage

 ```1 2 3 4``` ```ssa2d(x=33, p0=runif(4, max=0.8), p1=colMeans(matrix(p0[c( 1,3, 2,3, 1,4, 2,4)], nrow=2))/2, set=(x^2+1)/2, all=TRUE, shape=c(1,1)) ```

## Arguments

 `x` a linear dimension of 2D square percolation lattice. `p0` a vector of relative fractions `(0

## Details

The percolation is simulated on 2D square lattice with uniformly weighted sites `acc` and the vectors `p0` and `p1`, distributed over the lattice directions, and their combinations.

The anisotropic cluster is formed from the accessible sites connected with the initial subset `set`, and depends on the direction in 2D square lattice.

To form the cluster the condition `acc[set+eN[n]]<pN[n]` is iteratively tested for sites of the Moore (1,d)-neighborhood `eN` for the current cluster perimeter `set`, where `eN` is equal to `e0` or `e1` vector; `pN` is equal to `p0` or `p1` vector; `n` is equal to direction in 2D square lattice.

Moore (1,d)-neighborhood on 2D square lattice consists of sites, at least one coordinate of which is different from the current site by one: `e=c(e0,e1)`, where
`e0=c(-1,` `1,` `-x,` `x)`;
`e1=colSums(matrix(e0[c(1,3,` `2,3,` `1,4,` `2,4)], nrow=2))`.

Minkowski distance between sites `a` and `b` depends on the exponent `d`:
`rhoM <- function(a, b, d=1)`
`if (is.infinite(d)) return(apply(abs(b-a), 2, max))`
`else return(apply(abs(b-a)^d, 2, sum)^(1/d))`.

Minkowski distance for sites from `e1` subset with the exponent `d=1` is equal to `rhoMe1=2`.

Forming cluster ends with the exhaustion of accessible sites in Moore (1,d)-neighborhood of the current cluster perimeter.

## Value

 `acc` an accessibility matrix for 2D square percolation lattice: if `acc[e]

## Author(s)

Pavel V. Moskalev

## References

 Moskalev, P.V. Percolation modeling of porous structures. Moscow: URSS, 2018. 240 pp; in Russian.

 ```1 2 3 4 5``` ```set.seed(20120507) x <- y <- seq(33) image(x, y, ssa2d(), zlim=c(0,2), main="Anisotropic (1,1)-cluster") abline(h=17, lty=2); abline(v=17, lty=2) ```