Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem
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Abstract:
We investigate criteria for circle packing (CP) types of disk triangulation graphs embedded into simply connected domains in $\mathbb {C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum _{n=1}^\infty \frac {1}{\sum _{j=0}^{n-1} (k_j +6)} = \infty , \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum _{v \in B_n} (\deg v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum _{n=1}^\infty \frac {1}{\sum _{j=0}^{n-1} (k_j +6)+\sum _{j=0}^{n} (k_j +6)} = \infty . \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998.References
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Additional Information
- Byung-Geun Oh
- Affiliation: Department of Mathematics Education, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, South Korea
- MR Author ID: 766393
- ORCID: 0000-0001-5777-8460
- Email: bgoh@hanyang.ac.kr
- Received by editor(s): April 22, 2020
- Received by editor(s) in revised form: January 31, 2021
- Published electronically: December 3, 2021
- Additional Notes: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03034665).
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 753-797
- MSC (2020): Primary 52C15, 05B40, 05C10
- DOI: https://doi.org/10.1090/tran/8503
- MathSciNet review: 4369235