Some criteria for circle packing types and combinatorial Gauss-Bonnet Theorem
- Authors
- Oh, Byung Geun
- Issue Date
- Feb-2022
- Publisher
- AMER MATHEMATICAL SOC
- Citation
- TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.375, no.2, pp.753 - 797
- Indexed
- SCIE
SCOPUS
- Journal Title
- TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
- Volume
- 375
- Number
- 2
- Start Page
- 753
- End Page
- 797
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/139682
- DOI
- 10.1090/tran/8503
- ISSN
- 0002-9947
- Abstract
- We investigate criteria for circle packing (CP) types of disk triangulation graphs embedded into simply connected domains in 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
Sigma(infinity)(n=1) 1/Sigma(n-1)(j=0) (k(j) + 6) = infinity,
where k(n) is the degree excess sequence defined by
k(n) = Sigma(v)(is an element of Bn) (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
Sigma(infinity)(n=1) 1/Sigma(n-1)(j=0) (k(j) + 6) + Sigma(n)(j=0) (k(j) + 6) = infinity.
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.
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