Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2011-11-04
Physics
Condensed Matter
Disordered Systems and Neural Networks
Scientific paper
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 critical polynomials can be defined on any periodic lattice. In general, 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 the critical polynomial can be viewed as a graph invariant like the chromatic and Tutte polynomials. Like these, the critical polynomial is computed on a finite graph and may be found using the deletion-contraction algorithm. This allows efficient 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 \times 10^{-7}.
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