Transfer matrix computation of generalised 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 $p_c(4,8^2) = 0.676 803 329 ...$, $p_c(\mathrm{kagome}) = 0.524 404 998 ...$, $p_c(3,12^2) = 0.740 420 798 ...$, 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.