Circle-packing connections with random walks and a finite volume method

 Access to full text

 Full (PDF)

1. [BeSc] I. BENJAMINI and O. SCHRAMM. - Harmonic fonctions on planar and almost planar graphs and manifolds, via circlepackings. Invent. Math. 126 ( 1996), 565-587. Zbl0868.31008
2. [BoSt] P. L. BOWERS and K. STEPHENSON. - A branched Andreev-Thurston theorem for circle packings on the sphere, Proc. London Math. Soc. 73( 1996), 185-215. Zbl0856.51012
3. [BR] R. E. BANK and D. J. ROSÉ. - Some error estimates for the box method, SIAM J. Numer. Anal. 24 ( 1987), 777-787. Zbl0634.65105
4. [BSt1] A. F. BEARDON and K. STEPHENSON. - The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 ( 1990), 1383-1425. Zbl0797.30008MR1087197
5. [BSt2] A. F. BEARDON and K. STEPHENSON.- The Schwarz-Pick lemma for circle packings, 111. J. Math. 141 ( 1991), 577-606. Zbl0753.30016MR1115988
6. [C] Z. CAL - On the unite volume element method, Numer. Math. 58 ( 1991), 713-735. Zbl0731.65093MR1090257
7. [CdV] Y. COLIN DE VERDIÈRE. - Un principe variationnel pour les empilements de cercles, Invent. Math. 104 ( 1991), 655-669. Zbl0745.52010MR1106755
8. [CdVMa] Y. COLIN DE VERDIÈRE and F. MATHÉUS. - Empilements de cercles et approximations conformes, Actes de la Table Ronde de Géométrie Riemannienne en l'honneur de Marcel Berger, A. L. Besse (éditeur), Collection SMF Séminaires et Congrès 1 ( 1996), 253-272. Zbl0932.52004MR1427761
9. [CMM] Z. CAI, J. MANDEL and S. MCCORMICK. - The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. 28 ( 1991), 392-402. Zbl0729.65086MR1087511
10. [De] D. R. DEBAUN.- L2-cohomology of noncompact surfaces, Proc. Amer. Math. Soc. 284 ( 1984), 543-565. Zbl0511.58002MR743732
11. [DoSn] P. DOYLE and J. SNELL. - Random walks and electric networks, Carus Mathematica! Monographs, 22, MAA, 1984. Zbl0583.60065MR920811
12. [D1] T. DUBEJKO. - Branched circle packings and discrete Blaschke products, Trans. Amer. Math. Soc. 347 ( 1995). 4073-1103. Zbl0849.30028MR1308008
13. [D2] T. DUBEJKO. - Recurrent random walks, Liouville's theorem, and circle packing, Math. Proc. Cambridge Philos. Soc. 121 ( 1997), 531-546. Zbl0888.30005MR1434659
14. [D3] T. DUBEJKO. - Random walks on circle packings, (Math Sci Res Inst preprint #066-95), Contemp. Math., (to appear). Zbl0898.60072MR1476986
15. [D4] T. DUBEJKO. - Discrete solutions of Dirichlet problems, finite volumes, and circle packings, preprint. Zbl0945.30004MR1692623
16. [ECH] R. EYMARD, T. GALLOUËT and R. HERBIN. - Finite Volume Methods, in preparation for the Handbook of Numerical Analysis, Ph. Ciarlet and J. L. Lions eds. Zbl0981.65095
17. [Ha] W. HACKBUSCH. - On first and second order box schemes, Computing 41 ( 1989), 277-296. Zbl0649.65052MR993825
18. [Hn] B. HEINRICH. - Finite difference methods on irregular networks, Birkhäuser-Verlag, 1987. Zbl0623.65096MR875416
19. [HSc1] Z.-X. HE and O. SCHRAMM. - Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 ( 1993). 369-406. Zbl0777.30002MR1207210
20. [HSc2] Z.-X. HE and O. SCHRAMM. - Hyperbolic and parabolic packings, Discrete and Comput. Geom. 14 (95). 123-149. Zbl0830.52010MR1331923
21. [HSc3] Z.-X. HE and O. SCHRAMM. - On convergence of circle packings to the Riemann map, Invent. Math. 125 ( 1996), 285-305. Zbl0868.30010MR1395721
22. [HR] Z.-X. HE and B. RODIN. - Convergence of circle packings of finite valence to Riemann mappings, Comm. in Analysis and Geometry 1 ( 1993), 31-41. Zbl0777.30003MR1230272
23. [Ma] F. MATHÉUS. - Empilements de cercles et discrétisation quasiconforme: comportement asymptotique des rayons. preprint. Zbl0930.30011MR1692619
24. [Mc] G. MCCAUGHAN. - A recurrence/transience resuit for circle packings, Proc. Amer. Math. Soc, (to appear). Zbl0912.30002MR1327026
25. [RS] B. RODIN and D. SULLIVAN. - The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 ( 1987). 349-360. Zbl0694.30006MR906396
26. [S] P. M. SOARDI. - Potential theory on infinite networks, Lecture notes in mathematics, 1590, Springer-Verlag, 1994. Zbl0818.31001MR1324344
27. [St1] K. STEPHENSON. - Circle packings in the approximation of conformal mappings, Bull. Amer. Math. Soc. (Research Announcements) 23, (2) ( 1990), 407-115. Zbl0714.30011MR1049434
28. [St2] K. STEPHENSON. - A probabilistic proof of Thurston's conjecture on circle packings, Rend. Sem. Mat. Fis. Milano, (to appear). Zbl0997.60504MR1639851
29. [St3] K. STEPHENSON. - CirclePack (software), http://www.math.utk.edu/~kens. 
30. [Th1] W.P. THURSTON. - The Geometry and Topology of 3-ManifoIds, Princeton University Notes, Princeton University Press, 1980. 
31. [Th2] W.P. THURSTON. - The finite Riemann mapping theorem, Invited talk, An International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture, March 1985. 
32. [W] W. WOESS. - Random walks on infinite graphs and groups - a survey on selected topics, Bull. London Math. Soc. 26 ( 1994), 1-60. Zbl0830.60061MR1246471