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Electrical response matrix of a regular  -gon
Author(s):
Nathaniel
D.
Blair-Stahn;
David
B.
Wilson
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2015-2025.
MSC (2000):
Primary 31A25;
Secondary 30C20, 82B20, 05C05
Posted:
January 21, 2009
MathSciNet review:
2480283
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Additional information
Abstract:
Consider a unit-resistive plate in the shape of a regular polygon with  sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular  -gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests.
References:
-
- [Car92]
- John L. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), no. 4, L201-L206. MR 1151081 (92m:82048)
- [CdV98]
- Yves Colin de Verdière, Spectres de graphes, Cours Spécialisés [Specialized Courses], vol. 4, Société Mathématique de France, Paris, 1998. MR 1652692 (99k:05108)
- [CIM98]
- E. B. Curtis, D. Ingerman, and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl. 283 (1998), no. 1-3, 115-150. MR 1657214 (99k:05096)
- [CS04]
- Gabriel D. Carroll and David Speyer, The cube recurrence, Electron. J. Combin. 11 (2004), no. 1, Research Paper 73, 31 pp.
MR 2097339 (2005f:05007) - [Dub06]
- Julien Dubédat, Euler integrals for commuting SLEs, J. Stat. Phys. 123 (2006), no. 6, 1183-1218.
MR 2253875 (2007g:82027) - [FK72]
- C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica 57 (1972), 536-564. MR 0359655 (50:12107)
- [Gam01]
- Theodore W. Gamelin, Complex Analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. MR 1830078 (2002h:30001)
- [GM05]
- John B. Garnett and Donald E. Marshall, Harmonic Measure, New Mathematical Monographs, vol. 2, Cambridge University Press, 2005. MR 2150803 (2006g:31002)
- [IM98]
- David Ingerman and James A. Morrow, On a characterization of the kernel of the Dirichlet-to-Neumann map for a planar region, SIAM J. Math. Anal. 29 (1998), no. 1, 106-115 (electronic). MR 1617177 (99c:35258)
- [Kir90]
- Gustav Kirchhoff, Beweis der Existenz des Potentials das an der Grenze des betrachteten Raumes gegebene Werthe hat für den Fall dass diese Grenze eine überall convexe Fläche ist, Acta Math. 14 (1890), no. 1, 179-183. MR 1554794
- [KW06]
- Richard W. Kenyon and David B. Wilson, Boundary partitions in trees and dimers, 2006. To appear in Trans. Amer. Math. Soc. http://arxiv.org/abs/math.PR/0608422
- [Smi01]
- Stanislav Smirnov, Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239-244. MR 1851632 (2002f:60193)
- [Zif95]
- Robert M. Ziff, Proof of crossing formula for
 D percolation, J. Phys. A 28 (1995), no. 22, 6479-6480. MR 1379947 (96m:82030)
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Additional Information:
Nathaniel
D.
Blair-Stahn
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
David
B.
Wilson
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, Washington 98052
DOI:
10.1090/S0002-9939-09-09734-2
PII:
S 0002-9939(09)09734-2
Received by editor(s):
April 3, 2007,
Received by editor(s) in revised form:
May 1, 2008
Posted:
January 21, 2009
Communicated by:
Edward C. Waymire
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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