Consider $n$ points arbitrarily located on the plane. Consider a random graph $G$ drawn from $G(n, \frac12)$ on these points (i.e. the Erdos-Renyi random graph where every edge is selected with probability $\frac12$).

What is known about the geometry of the minimum spanning tree of such a graph? I am interested in pointers to **any** literature on this, but something like the following might be a concrete example:

Thm.With high probability, the Minimum Spanning Tree has weight within a factor of $\alpha$ of the MST on the complete graph on the same points.