Diagramatic Analysis of Colored Percolation in Three Dimensions

We consider a three dimesional version of colored percolation, examined previously in a 2D context (Sumanta Kundu and S. S. Manna, Phys. Rev. E 95, 052124 [2017]) on a cubic lattice with site occupation probability $p$ and assigned one of $N$ different colors. Neighboring sites, when occupied by dissimilar colors are deemed members of a connected cluster. We obtain high order series expansions for relevant observables with a technique in the vein of that used by J. L. Martin, which we analyze with Pade Approximants. In this manner, we obtain critical indices such as the percolation threshold $p_{c}$ and critical exponents. We confirm these results with large scale Monte Carlo simulations; we also assess whether the 3D colored percolation model belongs to the same universality class as other three dimensional percolation models.