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

TL;DR

This paper studies bond percolation on complex fractals like the non-p.c.f. Sierpinski gasket, the hexacarpet, and barycentrically subdivided triangles, establishing bounds on critical probabilities and revealing phase transition properties.

Contribution

It introduces bounds on percolation thresholds for these fractals using diamond fractals and uncovers duality and phase transition phenomena in non-p.c.f. fractal graphs.

Findings

Critical probability bounded above by 0.282 for the studied fractals.

Duality properties between gasket and hexacarpet fractals.

Existence of non-trivial phase transition on all three graphs.

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.