Layered circlepackings and the type problem
HTML articles powered by AMS MathViewer
- by Ryan Siders PDF
- Proc. Amer. Math. Soc. 126 (1998), 3071-3074 Request permission
Abstract:
We study the geometric type of a surface packed with circles. For circles packed in concentric layers of uniform degree, the circlepacking is specified by this sequence of degrees. We write an infinite sum whose convergence discerns the geometric type: if $h_i$ layers of degree $6$ follow the $i$th layer of degree $7$, and the $i$th layer of degree $7$ has $c_i$ circles, then $\sum \log (h_i)/c_i$ converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree $>6$.References
- Peter G. Doyle, On deciding whether a surface is parabolic or hyperbolic, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 41–48. MR 954627, DOI 10.1090/conm/073/954627
- Peter G. Doyle and J. Laurie Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR 920811, DOI 10.5948/UPO9781614440222
- He, Z X, and Schramm, O, Hyperbolic and Parabolic Packings, preprint.
- Russell Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), no. 3, 931–958. MR 1062053
- McCaughan, G, Transience and Recurrence, Proceedings of the AMS, to appear.
Additional Information
- Ryan Siders
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Email: rcsiders@math.princeton.edu
- Received by editor(s): November 28, 1995
- Received by editor(s) in revised form: February 28, 1997
- Additional Notes: This work was done under Dr. Phil Bowers of Florida State University during FSU’s 1994 Research Experience for Undergraduates. Dr. Bowers was an inspiring mentor. I will treasure what I learned from our conversations. The program was sponsored by the NSF
- Communicated by: James West
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3071-3074
- MSC (1991): Primary 52C15
- DOI: https://doi.org/10.1090/S0002-9939-98-04472-4
- MathSciNet review: 1459150