February  2019, 12(1): 27-42. doi: 10.3934/dcdss.2019002

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

URL: https://djkelleher.wordpress.com/
URL: http://www.math.uconn.edu/~panzo/
URL: http://www.math.uconn.edu/~teplyaev/

Received  November 2016 Revised  October 2017 Published  July 2018

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

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.

Citation: Daniel J. Kelleher, Hugo Panzo, Antoni Brzoska, Alexander Teplyaev. Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 27-42. doi: 10.3934/dcdss.2019002
References:
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show all references

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L. AmbrosioM. Erbar and G. Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, Nonlinear Anal., 137 (2016), 77-134.  doi: 10.1016/j.na.2015.12.006.  Google Scholar

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M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. London Ser. A, 431 (1990), 345-360.  doi: 10.1098/rspa.1990.0135.  Google Scholar

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[6]

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[7]

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[10]

F. Baudoin and D. J. Kelleher, Differential one-forms on dirichlet spaces and bakry-emery estimates on metric graphs, arXiv: 1604.02520, Transactions of the AMS, to appear. doi: 10.1090/tran/7362.  Google Scholar

[11]

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[12]

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[13]

R. BellC.-W. Ho and R. S. Strichartz, Energy measures of harmonic functions on the Sierpiński gasket, Indiana Univ. Math. J., 63 (2014), 831-868.  doi: 10.1512/iumj.2014.63.5256.  Google Scholar

[14]

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[15]

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[19]

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[20]

P. Diaconis and L. Miclo, On barycentric subdivision, Combin. Probab. Comput., 20 (2011), 213-237.  doi: 10.1017/S0963548310000441.  Google Scholar

[21]

P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984.  Google Scholar

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M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19 of de Gruyter Studies in Mathematics, extended edition, Walter de Gruyter & Co., Berlin, 2011.  Google Scholar

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[24]

A. Grigor'yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math., 66 (2014), 641-699.  doi: 10.4153/CJM-2012-061-5.  Google Scholar

[25]

A. Grigor'yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab., 40 (2012), 1212-1284.  doi: 10.1214/11-AOP645.  Google Scholar

[26]

B. M. HamblyV. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc. (2), 74 (2006), 93-112.  doi: 10.1112/S002461070602312X.  Google Scholar

[27]

K. E. HareB. A. SteinhurstA. Teplyaev and D. Zhou, Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals, Math. Res. Lett., 19 (2012), 537-553.  doi: 10.4310/MRL.2012.v19.n3.a3.  Google Scholar

[28]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[29]

J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, vol. 27 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2015, An approach based on upper gradients. doi: 10.1017/CBO9781316135914.  Google Scholar

[30]

M. HinzD. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 408 (2012), 303-322, 329–330.   Google Scholar

[31]

M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), 299-317.   Google Scholar

[32]

M. HinzD. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms, J. Noncommut. Geom., 9 (2015), 359-390.  doi: 10.4171/JNCG/195.  Google Scholar

[33]

M. Hinz, M. R. Lacia, A. Teplyaev and P. Vernole, Fractal snowflake domain diffusion with boundary and interior drifts, J. Math. Anal. Appl., 457 (2018), 672–693, arXiv: 1605.06785. doi: 10.1016/j.jmaa.2017.07.065.  Google Scholar

[34]

M. HinzM. Röckner and A. Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406.  doi: 10.1016/j.spa.2013.06.009.  Google Scholar

[35]

M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854.  doi: 10.1016/j.jfa.2013.07.021.  Google Scholar

[36]

M. IonescuL. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169.  doi: 10.1016/j.jfa.2012.05.021.  Google Scholar

[37]

V. A. Kaimanovich, "Münchhausen trick" and amenability of self-similar groups, Internat. J. Algebra Comput., 15 (2005), 907-937.  doi: 10.1142/S0218196705002694.  Google Scholar

[38]

N. Kajino, Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, Potential Anal., 36 (2012), 67-115.  doi: 10.1007/s11118-011-9221-5.  Google Scholar

[39]

N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpiński gasket, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics, vol. 600 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, 91–133. doi: 10.1090/conm/600/11932.  Google Scholar

[40]

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D. J. Kelleher, B. A. Steinhurst and C. -M. M. Wong, From self-similar structures to selfsimilar groups, Internat. J. Algebra Comput., 22 (2012), 1250056, 16pp. doi: 10.1142/S0218196712500567.  Google Scholar

[42]

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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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