Transfer matrix computation of generalized critical polynomials in percolation

Percolation thresholds have recently been studied by means of a graph polynomial P-B(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P-B(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P-B(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P-B(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 8(2)), kagome, and (3, 12(2)) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p(c)(4, 8(2)) = 0.676 803 329 ..., p(c)(kagome) = 0.524 404 998 ..., p(c)(3, 12(2)) = 0.740 420 798 ..., comparable to the best simulation results. We also show that the alternative definition of P-B(p) can be applied to study site percolation problems.