Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions

TL;DR

This paper establishes resistance estimates for random walks on barycentrically subdivided triangles, revealing different scaling behaviors depending on the walk's jump locations, and discusses implications for spectral dimension and self-similar forms.

Contribution

It provides the first resistance estimates for random walks on barycentric subdivisions, highlighting differences in behavior based on jump points and suggesting complex limiting behaviors.

Findings

Resistance scales as a power of a constant .06 for walks between triangle centers.

Resistance scales as a power of a constant /3 for walks between triangle corners.

Spectral dimension estimates range between 1.63 and 2.38 depending on walk type.

Abstract

We prove Barlow--Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant $\rho$ which is theoretically estimated to be in the interval $5/4\leqslant\rho\leqslant3/2$, with a numerical estimate $\rho\approx1.306$. This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S\approx1.74$. On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant $\rho^T=1/\rho$, which is theoretically estimated to be in the interval $2/3\leqslant\rho^T\leqslant4/5$. This corresponds to the spectral dimension between 2.28 and 2.38. The difference between $\rho$ and $\rho^T$ implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.