fssi2d {SPSL}

Frequency of Sites on a Square Isotropic 2D lattice with (1,d)-neighborhood

Description

fssi2d() function calculates the relative frequency distribution of isotropic clusters on 2D square lattice with Moore (1,d)-neighborhood.

Usage

fssi2d(n=1000, x=33, p0=0.5, p1=p0/2, 
       set=(x^2+1)/2, all=TRUE, shape=c(1,1))

Arguments

n

a sample size.

x

a linear dimension of 2D square percolation lattice.

p0

a relative fraction (0<p0)&(p0<1) of accessible sites (occupation probability) for percolation lattice.

p1

p0 value, weighted by Minkowski distance: p1=p0/rhoMe1.

set

a vector of linear indexes of a starting sites subset.

all

logical; if all=TRUE, mark all sites from a starting subset; if all=FALSE, mark only accessible sites from a starting subset.

shape

a vector with two shape parameters of beta-distributed random variables, weighting the percolation lattice sites.

Details

The percolation is simulated on 2D square lattice with uniformly weighted sites and the constant parameters p0 and p1.

The isotropic cluster is formed from the accessible sites connected with initial sites subset set.

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, -x^2, x^2);
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.

Each element of the matrix frq is equal to the relative frequency with which the 2D square lattice site belongs to a cluster sample of size n.

Value

rfq

a 2D matrix of relative sampling frequencies for sites of the percolation lattice.

Author(s)

Pavel V. Moskalev <[e-mail]>

References

[1] Moskalev, P.V. Percolation modeling of porous structures. Moscow: URSS, 2018. 240 pp; in Russian.
[2] Moskalev, P.V. (2014) Estimates of threshold and strength of percolation clusters on square lattices with (1,d)-neighborhood. Computer Research and Modeling, Vol.6, No.3, pp.405–414; in Russian.

See Also

ssi2d, fssi3d, fssi20, fssi30, fssa2d, fssa3d

Examples

x <- y <- seq(33)
image(x, y, rfq <- fssi2d(n=200), cex.main=1,
main="Frequencies of isotropic (1,1)-clusters")
contour(x, y, rfq, levels=seq(.2,.3,.05), add=TRUE)
abline(h=17, lty=2); abline(v=17, lty=2)