Bond percolation on a non-p.c.f. Sierpi\'nski Gasket, iterated barycentric subdivision of a triangle, and Hexacarpet

Derek Lougee; Benjamin Steinhurst

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arXiv · 1302.5436

Bond percolation on a non-p.c.f. Sierpi\'nski Gasket, iterated barycentric subdivision of a triangle, and Hexacarpet

Derek Lougee·Benjamin Steinhurst

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Abstract

We investigate bond percolation on the iterated barycentric subdivision of a triangle, the hexacarpet, and the non-p.c.f. Sierpinski gasket. With the use of the diamond fractal, we are able to bound the critical probability of percolation on the non-p.c.f. gasket and the iterated barycentric subdivision of a triangle from above by 0.282. We then show how both the gasket and hexacarpet fractals are related via the iterated barycentric subdivisions of a triangle: the two spaces exhibit duality properties although they are not themselves dual graphs. Finally we show the existence of a non-trivial phase transition on all three graphs.

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Derek Lougee, Benjamin Steinhurst. 2013-02-21. Bond percolation on a non-p.c.f. Sierpi\'nski Gasket, iterated barycentric subdivision of a triangle, and Hexacarpet. https://arxiv.org/abs/1302.5436

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