Counting Clusters on a Finite Grid

We consider p = 1/2 site percolation on finite rectangles in the 2D square lattice. For n,m ∈ Z, let e(m,n) be the expected number of clusters connected monochromatic components over all two-colorings of [0,m)× [0, n)∩Z, where points are connected orthogonally. We determine the values e(m,n) explicitly for small values of m, and demonstrate connections between the expected number of clusters, expected cluster size, monochromatic cycles, and other plane animals. We show that the limit λ = lim m,n→∞ e(m,n) mn exists, and bound the constant by 29 448 ≤ λ ≤ 1 12 . We also give similar results for some related lattices, such as the 2D torus and hexagonal lattices, and extend the theory to some more general graphs.