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

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

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

Daniel J. Kelleher ^1, , Hugo Panzo ^2, , Antoni Brzoska ^2, and Alexander Teplyaev ^2,

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.

Keywords: Repeated barycentric subdivision, random walk, Sierpinski carpet, Strichartz hexacarpet, Barlow–Bass resistance estimate, spectral dimension, self-similar Dirichlet form.

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

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

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

References:

[1]	L. Ambrosio, M. 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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[3]	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
[4]	M. T. Barlow, R. F. Bass and J. D. Sherwood, Resistance and spectral dimension of Sierpiński carpets, J. Phys. A, 23 (1990), L253-L258. doi: 10.1088/0305-4470/23/6/004. Google Scholar
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[12]	M. Begue, D. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet, Exp. Math., 21 (2012), 402-417. doi: 10.1080/10586458.2012.715542. Google Scholar
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[14]	J. Bello, Y. Li and R. S. Strichartz, Hodge–de Rham theory of K-forms on carpet type fractals, in Excursions in Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 3 (2015), 23–62. Google Scholar
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[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. Hambly, V. 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. Hare, B. A. Steinhurst, A. 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
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[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. Hinz, D. 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. Hinz, D. 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. Hinz, M. 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. Ionescu, L. 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]	C. J. Kauffman, R. M. Kesler, A. G. Parshall, E. A. Stamey and B. A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys., 53(2012), 042102, 18pp. doi: 10.1063/1.3702099. Google Scholar
[41]	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]	M. Keller, D. Lenz and R. K. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Math. Z., 274 (2013), 905-932. doi: 10.1007/s00209-012-1101-1. Google Scholar
[43]	J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755. doi: 10.2307/2154402. Google Scholar
[44]	J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar
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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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Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

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

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