Site percolation on the square lattice isn?t self-dual since there can be ?checkerboard? states where neither path is possible,
I could be wrong, but I don't think that's what anyone is talking about.
Right, sorry for my misunderstanding. I think the issue with the Bollobas and Riordan paper is this. Let R_n(p) be the probability that there is a path of occupied bonds from North to South on an n-by-n square, if each bond is occupied with probability p. Duality shows that there is such a path if and only if there is not a path of non-occupied bonds from East to West. Therefore, R_n(p) = 1-R_n(1-p) . I think the challenge is showing that there is a sharp threshold when n tends to infinity, i.e., a p_c such that R_n(p) tends to 0 if p<p_c and 1 if p>p_c. Once we know this, then duality tells us that p_c=1/2. But proving that it’s a phase transition with this zero-one phenomenon is the trick.
and the threshold is 0.592746? > 1/2.
Where does that number come from?
There are lots of ways to estimate it from Monte Carlo experiments and by extrapolating from exact computations on finite squares. Here’s a nice page: https://en.wikipedia.org/wiki/Percolation_threshold <https://en.wikipedia.org/wiki/Percolation_threshold> There are a few lattices where a local transformation gives the exact threshold. For instance, for bond percolation on the honeycomb lattice, p_c is a root of 1 + p^3 - 3 p^2 = 0 because a cool thing called the star-triangle relation can be used to jump between the honeycomb and its dual the triangular lattice. Cris
Allan Wechsler <acwacw@gmail.com> wrote:
Keith's confusion, which I share, is raised by the wording in the Bollobas & Riordan paper which he quotes upthread. It seems to Keith and me that pH = 1/2 follows instantly from self-duality, but Bollobas & Riordan say that the proof is deep and non-trivial. I know Bollobas's name -- these are no lightweights. It follows that Keith and I are missing some complication to the question.
All I can think of is that maybe what I describe above as obvious is one of those things that's very difficult to formally prove. I certainly wouldn't know how to begin to do so. It's one of those things that would seem to not need a proof. I guess that means I'll never be a real mathematician. See https://xkcd.com/2042/
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