32 research outputs found
Exact bond percolation thresholds in two dimensions
Recent work in percolation has led to exact solutions for the site and bond
critical thresholds of many new lattices. Here we show how these results can be
extended to other classes of graphs, significantly increasing the number and
variety of solved problems. Any graph that can be decomposed into a certain
arrangement of triangles, which we call self-dual, gives a class of lattices
whose percolation thresholds can be found exactly by a recently introduced
triangle-triangle transformation. We use this method to generalize Wierman's
solution of the bow-tie lattice to yield several new solutions. We also give
another example of a self-dual arrangement of triangles that leads to a further
class of solvable problems. There are certainly many more such classes.Comment: Accepted for publication in J. Phys
The percolation critical polynomial as a graph invariant
Every lattice for which the bond percolation critical probability can be
found exactly possesses a critical polynomial, with the root in [0,1] providing
the threshold. Recent work has demonstrated that this polynomial may be
generalized through a definition that can be applied on any periodic lattice.
The polynomial depends on the lattice and on its decomposition into identical
finite subgraphs, but once these are specified, the polynomial is essentially
unique. On lattices for which the exact percolation threshold is unknown, the
polynomials provide approximations for the critical probability with the
estimates appearing to converge to the exact answer with increasing subgraph
size. In this paper, I show how this generalized critical polynomial can be
viewed as a graph invariant, similar to the Tutte polynomial. In particular,
the critical polynomial is computed on a finite graph and may be found using
the recursive deletion-contraction algorithm. This allows calculation on a
computer, and I present such results for the kagome lattice using subgraphs of
up to 36 bonds. For one of these, I find the prediction p_c=0.52440572...,
which differs from the numerical value, p_c=0.52440503(5), by only 6.9 x
10^{-7}
Critical manifold of the kagome-lattice Potts model
Any two-dimensional infinite regular lattice G can be produced by tiling the
plane with a finite subgraph B of G; we call B a basis of G. We introduce a
two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in
G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the
critical manifold of the q-state Potts model, with coupling v = exp(K)-1,
defined on G. This curve predicts the phase diagram both in the ferromagnetic
(v>0) and antiferromagnetic (v<0) regions. For larger bases B the
approximations become increasingly accurate, and we conjecture that P_B(q,v) =
0 provides the exact critical manifold in the limit of infinite B. Furthermore,
for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises
for any choice of B: the zero set of the recurrent factor then provides the
exact critical manifold. In this sense, the computation of P_B(q,v) can be used
to detect exact solvability of the Potts model on G.
We illustrate the method for the square lattice, where the Potts model has
been exactly solved, and the kagome lattice, where it has not. For the square
lattice we correctly reproduce the known phase diagram, including the
antiferromagnetic transition and the singularities in the Berker-Kadanoff
phase. For the kagome lattice, taking the smallest basis with six edges we
recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases
provide successive improvements on this formula, giving a natural extension of
Wu's approach. The polynomial predictions are in excellent agreement with
numerical computations. For v>0 the accuracy of the predicted critical coupling
v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to
10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).Comment: 31 pages, 12 figure