Site percolation on the square lattice isn’t self-dual since there can be “checkerboard” states where neither path is possible, and the threshold is 0.592746… > 1/2. But bond percolation on the square lattice is self-dual, and so is site percolation on the triangular lattice (see Hex). Maybe we’re not talking about the same things? Cris On Apr 13, 2019, at 8:58 AM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Allan Wechsler <acwacw@gmail.com> wrote:
Just in case anybody doesn't know: questions like this have been studied at least since the 1950s, under the rubric of "percolation theory". The seminal work appears to be Broadbent & Hammersley 1957. See the Wikipedia article on percolation theory for the complete reference.
Thanks. Wikipedia led me, indirectly, to https://arxiv.org/pdf/math/0410359.pdf which says:
The problem of investigating pH in a variety of contexts was posed by Broadbent and Hammersley [7] in 1957. Hammersley [11, 12, 13] proved general results implying in particular that 0.35 < pH(Z2) < 0.65. The first major progress was due to Harris [14], who proved in 1960 that pH(Z2) >= 1/2. His proof makes use of the "self-duality" of Z^2, and is highly non-trivial. For many years it was believed that pH = 1/2 for bond percolation in Z^2; see, for example, Sykes and Essam [22]. However, it was only in 1980, twenty years after Harris proved that pH >= 1/2, that Kesten [16] proved this conjecture, following significant progress by Russo [20] and Seymour and Welsh [21].
I'm puzzled by this. I would think that knowing that it was self dual and that the odds of a solution were greater than or equal to one half would prove that the dual also has odds of a solution greater than or equal to one half. And since the dual obviously has a solution if and only if the original does not, it must follow that both are exactly equal to one half. What am I missing?
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