A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

John C. Wierman, Gaoran Yu, Theodore Huang

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In 1982, Tsallis derived a formula which proposed an exact value of 0.522372078... for the bond percolation threshold of the kagome lattice. We use the substitution method, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bond percolation model on the martini lattice. The bounds obtained are $0.522394<p_c(\mbox{kagome}) < 0.526750$, where the lower bound shows that the value conjectured by Tsallis is incorrect.

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percolation theory, set partition, stochastic ordering

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