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

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

    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 $G_2^T$

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

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

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

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

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

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

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

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

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