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A probabilistic proof of Thurston’s conjecture on circle packings. (English) Zbl 0997.60504

Thurston conjectured, and B. Rodin and D. Sullivan proved [J. Differ. Geom. 26, No. 2, 349-360 (1987; Zbl 0694.30006)], that circle packings can be used to approximate the conformal Riemann map from the unit disk to a simply connected domain in the plane. The current paper offers another proof of this result, one which is based on interpreting the flow of curvature in a circle packing as it deforms as the expected behavior of a random walk on a suitably tailored Markov chain. It is not assumed that the circle packings have the hexagonal combinatorics, but only that the interstices are triangular and that there is a finite upper bound on the ratio between any two radii in the packing.

MSC:

60G50 Sums of independent random variables; random walks
30C99 Geometric function theory

Citations:

Zbl 0694.30006
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References:

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