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 $\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show that $\mu_k\approx k^2$. Most of the paper is concerned with the case $k=2$. We show that $\m_2$ tends to an explicitly defined constant and that $\mu_2\approx 4.1704288\ldots$.