On the value of a random minimum weight steiner tree

Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1 + o(1))(k - 1)(logn - log k)/n when k = o(n) and n → ∞.