Consider the dual lattice, and prove that the critical probability is exactly p=1/2... Then consider placing hexagonal lily pads on the surface of a pond, and show that the critical probability for being able to cross the pond is again p=1/2. Hint: there are no draws in Hex. Cris On Feb 10, 2014, at 8:00 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Draw a maze on a two-dimensional n x n grid by erecting a wall between each pair of adjacent (ie, distance one) lattice points independently with probability p.
Eyeballing these things in Mathematica, it looks to me like if p < 1/2, there tends to be one "large component" that connects almost all the cells that are not walled off into "locally small" (say 1x1 or 1x2) walled gardens.
I'm sure I can't be the first to have considered something like this.
I'd welcome information about prior work.
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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