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

Daniel J. Kelleher; Hugo Panzo; Antoni Brzoska; Alexander Teplyaev

doi:10.3934/dcdss.2019002

Article Contents

2019, Volume 12, Issue 1: 27-42. Doi: 10.3934/dcdss.2019002

This issue Previous Article Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets Next Article Asymptotics for quasilinear obstacle problems in bad domains

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

1.
Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, USA
2.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

Received: October 31, 2016

Revised: September 30, 2017

Published: July 2018

Research supported in part by NSF grants DMS 1106982, 1262929, 1613025.

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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 $ρ$ which is theoretically estimated to be in the interval $5/4≤ρ≤3/2$ , with a numerical estimate $ρ≈1.306$ . This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S≈1.74$ . 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 $ρ^T = 1/ρ$ , which is theoretically estimated to be in the interval $2/3≤ρ^T≤4/5$ . This corresponds to the spectral dimension between 2.28 and 2.38. The difference between $ρ$ and $ρ^T$ 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.

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Mathematics Subject Classification: Primary: 60J35, 81Q35, 28A80; Secondary: 31C25, 31E05, 35K08.

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Figure 1. On the left: barycentric subdivision of a 2-simplex, the graphs $G_0^T$, $G_1^T$ and $G_2^T$. On the right: adjacency (dual) graph $G_2$, in bold, pictured together with the thin image of $G_2^T$

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Figure 2. On the left: the graph $G_4^T$ for barycentric subdivision of a 2-simplex. On the right: the adjacency (dual) graph $G_4$

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Figure 3. $A_{(2)}^{T/H}$ and $B_{(2)}^{T/H}$ on the hexagonal embedding of $G_2^{T/H}$

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Figure 4. $\tilde G^H_2$ and $\tilde G_2^T$ without the additional edges

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Figure 5. The transformation from the flow $I^n$ (left) to the flow $H_{02}^n$ (right)

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Figure 6. The function $u$, $v$ and $w$

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Figure 7. Gluing from $\widehat{G_1}$ to $\widehat{G_2}$

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Figure 8. Short-circuited graphs $\widetilde G^T_1$ and $\widetilde G^T_2$

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Figure 9. Left: Graph $\widetilde G_2^H$ with short circuits. Right: Non-p.c.f Sierpinski gasket

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