Transfer matrix computation of generalized critical polynomials in percolation

Abstract

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.

The three-state Potts antiferromagnet on plane quadrangulations

“…seven) quadrangulations of self-dual (resp. non-self-dual) type using several complementary techniques: MC simulations, TM computations, and the method of critical polynomials (CP) [28,35,36,70,71]. These numerical results agree, without exception, with conjecture 1.1.…”

“…The location of the phase transitions for the quadrangulations in figures 2 and 3 can also be studied by the method of critical polynomials [28,35,36,70,71]. These polynomials P B (q, v) can in principle be computed for any lattice generated by the tessellation of the plane by some finite basis B.…”

Percolation thresholds in hyperbolic lattices

“…In Fig. 7 we plot p site c ({P, Q}) − 1/(Q − 1) versus P on a semilog scale, and the straight lines indicate that scaling of the form (19) holds. We also compare p site c − 1/(Q − 1) for Q = 7 directly to our rigorous bounds in Fig.…”

Phase diagram of the triangular-lattice Potts antiferromagnet

“…We now discuss a completely different technique for determining the phase diagram: the method of CP. There is now a reasonably large body of work on this [38][39][40][41][42]77], so we provide only a brief overview here and refer the reader to the literature for details and justification of the method. The first step is to choose a finite graph, which we call the basis, that can be copied and translated to cover the entire lattice.…”