Planar Maps, Random Walks and Circle Packing by Asaf Nachmias

Author:Asaf Nachmias
Language: eng
Format: epub, pdf
ISBN: 9783030279684
Publisher: Springer International Publishing

(4.3)

Choose some v J ∈ ∂G J ∩ W ε(z) so that . For each j ≥ J choose v j ∈ ∂G j ∩ W ε(z) so that v j and v J are in the same connected component A j of the graph spanned on V j ∩ W ε(z). Since the circle of v j in P j touches we learn by (4.3) that the distance of the circle of v J in P j from is at most ε c for all j ≥ J. Since the circle corresponding to v J in P ∞ is the limit of its circles in P j we deduce that the distance of from is at most ε c. Hence the distance of z from is at most ε + ε c. Since ε was arbitrary we obtain that , as required. □

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