  Assign i.i.d. standard exponential edge weights to the edges of the complete
graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n
converges in the local weak sense (also called Aldous-Steele or
Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be
viewed as the component containing the root in the wired minimum spanning
forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov
process construction of M starting from the invasion percolation cluster on the
PWIT. We then show that M has cubic volume growth, up to lower order
fluctuations for which we provide explicit bounds. Our volume growth estimates
confirm recent predictions from the physics literature, and contrast with the
behaviour of invasion percolation on the PWIT and on regular trees, which
exhibit quadratic volume growth.
