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A probabilistic proof of Thurston's conjecture on circle packings

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Abstract

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.

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Authors and Affiliations

  1. University of Tennessee, Knoxvilie

    Kenneth Stephenson

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  1. Kenneth Stephenson
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Conferenza tenuta il 17 giugno 1996

The author gratefully aknowledges the support of the National Science Foundation and the Tennessee Science Alliance and the hospitality of the Seminario Matematico e Fisico di Milano.

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Stephenson, K. A probabilistic proof of Thurston's conjecture on circle packings. Seminario Mat. e. Fis. di Milano 66, 201–291 (1996). https://doi.org/10.1007/BF02925361

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