On edge disjoint spanning trees in a randomly weighted complete graph

Assume that the edges of the complete graph K_n are given independent uniform [0,1] edges weights. We consider the expected minimum total weight μ_k of k≥2 edge disjoint spanning trees. When k is large we show that μ_k≈k². Most of the paper is concerned with the case k=2. We show that $\m_2$ tends to an explicitly defined constant and that μ2≈4.1704288….