Critical volume fraction of percolation

Let us move to the percolation problem in the continuum spaces [31], This can be naturally done by first finding the invariants of the problem which do not depend on the details of the lattice structures but depend only on the space dimensions and symmetry, and then discard the lattices from which the problem started. 

For instance, the threshold value pdK s) of the site percolation depends on the lattice structure A, but when it is multiplied by the filling factor /(A), the product 

The critical volume fraction is therefore (pc = 1-00 for the one-dimensional lattice, and (pc = n ll) X 1/2 = 0.393 for two dimensions as calculated in the square lattice. Scher and Zallen [32] found pc = 0.44 0.02 in two dimensions d = l, while /c=0.154 0.005 in three dimensions d = h. It decreases with the space dimensions, because even if the cluster is percolated d dimensionally, it may not necessarily be percolated d 1 

Construction of the space filling factor / for the given lattice. 

As another example of the invariant, let us take the critical bond number 

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