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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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.

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M. Barlow and R. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673-744. doi: 10.4153/CJM-1999-031-4.

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M. BarlowR. F. BassT. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701.

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show all references

References:
[1]

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.

[2]

M. T. Barlow, Analysis on the Sierpinski carpet, in Analysis and Geometry of Metric Measure Spaces, vol. 56 of CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 2013, 27–53.

[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.

[4]

M. T. BarlowR. 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.

[5]

M. Barlow, Diffusions on fractals, in Lectures on Probability Theory and Statistics (SaintFlour, 1995), vol. 1690 of Lecture Notes in Math., Springer, Berlin, 1998, 1–121. doi: 10.1007/BFb0092537.

[6]

M. Barlow and R. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673-744. doi: 10.4153/CJM-1999-031-4.

[7]

M. BarlowR. F. BassT. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701.

[8]

L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25–118.

[9]

R. Bass, Diffusions on the Sierpinski carpet, in Trends in Probability and Related Analysis (Taipei, 1996), World Sci. Publ., River Edge, NJ, 1997, 1–34.

[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.

[11]

F. BauerM. Keller and R. K. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS), 17 (2015), 259-271. doi: 10.4171/JEMS/503.

[12]

M. BegueD. KelleherA. NelsonH. PanzoR. 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.

[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.

[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.

[15]

N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, vol. 14 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389.

[16]

Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton Univ. Press, 2012.

[17]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.

[18]

P. Diaconis and C. McMullen, Barycentric Subdivision Unpublished, 2008.

[19]

P. Diaconis and L. Miclo, On barycentric partitions, with simulations, https://hal.archives-ouvertes.fr/hal-00353842.

[20]

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

[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.

[22]

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.

[23]

R. Grigorchuk and V. Nekrashevych, Self-similar groups, operator algebras and Schur complement, J. Mod. Dyn., 1 (2007), 323-370. doi: 10.3934/jmd.2007.1.323.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[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.

[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.

[42]

M. KellerD. 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.

[43]

J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755. doi: 10.2307/2154402.

[44]

J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.

[45]

J. Kigami, Local Nash inequality and inhomogeneity of heat kernels, Proc. London Math. Soc. (3), 89 (2004), 525-544. doi: 10.1112/S0024611504014807.

[46]

J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc., 199 (2009), ⅷ+94pp. doi: 10.1090/memo/0932.

[47]

J. Kigami, Quasisymmetric modification of metrics on self-similar sets, in Geometry and Analysis of Fractals, vol. 88 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014,253–282. doi: 10.1007/978-3-662-43920-3_9.

[48]

O. Knill, The graph spectrum of barycentric refinements, arXiv: 1508.02027.

[49]

O. Knill, Universality for Barycentric subdivision, arXiv: 1509.06092.

[50]

S. Kusuoka and X. Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields, 93 (1992), 169-196. doi: 10.1007/BF01195228.

[51]

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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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