Nicolas Broutin

The scaling limit of the minimum spanning tree of a complete graph


We consider a complete graph weighted with iid uniform weights and build the minimum spanning tree $T_n$. The tree $T_n$ has attracted a lot of attention, but most informations known about its structure are local, even the famous result of Frieze saying that the total weight of converges to $\zeta(3)$. We are interested in the global structure of as $n\to\infty$, and consider it as a random metric space equipped with the graph distance $d_{T_n}$. We show that there exists a limit compact metric space $M$ such that $(T_n, n^{-1/3}d_{T_n})$ converges in distribution to $M$. The metric space is a continuum random tree, that we prove different from Aldous' CRT using arguments relying on its fractal dimension.

This is a joint work with L. Addario-Berry, C. Goldschmidt and G. Miermont.