caa a22 4500 477103855 CHVBK 20180405111548.0 cr unu---uuuuu 170330e19961201xx s 000 0 eng 10.1007/BF02925361 doi (NATIONALLICENCE)springer-10.1007/BF02925361 Stephenson Kenneth University of Tennessee, Knoxvilie aut A probabilistic proof of Thurston's conjecture on circle packings [Elektronische Daten] [Kenneth Stephenson] In 1985 William Thurston conjectured that one could use circle packings to approximate conformal mappings. This was confirmed by Burt Rodin and Dennis Sullivan with a proof which relied on the hexagonal nature of the packings involved. This paper provides a probabilistic proof which accomodates more general combinatorics by analysing the dynamics of invididual circle packings. One can use reversible Markov processes to model the movement of curvature and hyperbolic area among the circles of a packing as it undergoes adjustement, much as one can use them to model the movement of current in an electrical circuit. Each circle packing has a Markov process intimately coupled to its geometry; the crucial local rigidity of the packing then appears as a a Harnack inequality for discrete harmonic functions of the process. Birkhäuser-Verlag, 1998 Rendiconti del Seminario Matematico e Fisico di Milano Birkhäuser-Verlag 66/1(1996-12-01), 201-291 0370-7377 66:1<201 1996 66 32 https://doi.org/10.1007/BF02925361 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02925361 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Stephenson Kenneth University of Tennessee, Knoxvilie aut NATIONALLICENCE 773 0- Rendiconti del Seminario Matematico e Fisico di Milano Birkhäuser-Verlag 66/1(1996-12-01), 201-291 0370-7377 66:1<201 1996 66 32 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer