Random walks on barycentric subdivisions and the Strichartz hexacarpet

TL;DR

This paper explores the connection between random walks on barycentric subdivisions of triangles and a new fractal called the Strichartz hexacarpet, establishing convergence to a diffusion process with a specific spectral dimension.

Contribution

It introduces the Strichartz hexacarpet, proves convergence of barycentric subdivisions to a self-similar space, and provides numerical evidence of random walk convergence to a diffusion process.

Findings

Barycentric subdivisions converge to a fractal space of dimension approximately 2.58.

Random walks on subdivisions approximate a diffusion process on the hexacarpet.

Numerical estimates suggest the diffusion has a spectral dimension around 1.74.

Abstract

We investigate the relation between simple random walks on repeated barycentric subdivisions of a triangle and a self-similar fractal, Strichartz hexacarpet, which we introduce. We explore a graph approximation to the hexacarpet in order to establish a graph isomorphism between the hexacarpet approximations and Barycentric subdivisions of the triangle, and discuss various numerical calculations performed on the these graphs. We prove that equilateral barycentric subdivisions converge to a self-similar geodesic metric space of dimension log(6)/log(2), or about 2.58. Our numerical experiments give evidence to a conjecture that the simple random walks on the equilateral barycentric subdivisions converge to a continuous diffusion process on the Strichartz hexacarpet corresponding to a different spectral dimension (estimated numerically to be about 1.74).