Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph