In 2001 I. Benjamini and O. Schramm proved that the distributional limit of a random graph with bounded degree is almost surely recurrent with respect to a random walk. To prove this fact they devised an interesting lemma about finite point sets in the plane. Since then this same lemma has been used several times in different contexts. In particular, in joint work with S. Rohde, we used it to show that the uniform infinite planar triangulation is almost surely a parabolic Riemann surface. In further investigation by the speaker it has recently been found that the conclusion of this lemma holds in any doubling metric space. In fact, the metric doubling condition is equivalent to the property described in the lemma.
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