BOND PERCOLATION ON A NON-P.C.F. SIERPIŃSKI GASKET, ITERATED BARYCENTRIC SUBDIVISION OF A TRIANGLE, AND HEXACARPET

Abstract

We investigate bond percolation on the iterated barycentric subdivision of a triangle, the hexa-carpet, and the non-p.c.f. Sierpinski gasket. With the use of known results on the diamond fractal, we are able to bound the critical probability of bond 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.