On the connectivity of the Poisson process on fractals
  For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high probability (log n/n)^(1/d). 
Gady Kozma, Zvi Lotker and Gideon Stupp