Polychromatic percolation: Coexistence of percolating species in highly connected lattices

A generalization of percolation from a two-species (black-and-white) random process to a multispecies (polychromatic) process has been developed. The division of the chromatic composition field into regions in which different numbers of colors percolate for $C$ colors on a lattice with percolation threshold $p_c$ has been analyzed. A panchromatic regime (all species percolate) occurs when $C<\mathrm{}{p}_c^{-1\mathrm{}}$, occupying a fraction ${(1-C{p}_c)}^{C-1\mathrm{}}$ of the composition field. Since polychromatic percolation has greatest scope for highly connected low-$p_c$ lattices, these ideas have been applied to site-percolation processes on $d$-dimensional close-packed lattices, as well as $d=2\mathrm{}$ and $d=3\mathrm{}$ lattices with long-range interactions. Also the concept of a lattice-independent dimensional-invariant critical volume fraction for site percolation has been extended to $d=4\mathrm{}$ and $d=5\mathrm{}$, but is shown to fail for $d>\mathrm{}8\mathrm{}$. Possible applications of polychromatic percolation are briefly discussed.