THE RANDOM MINIMAL SPANNING TREE IN HIGH DIMENSIONS

For the minimal spanning tree on n independent uniform points in the d -dimensional unit cube, the proportionate number of points of degree k is known to converge to a limit alpha(k,d) as n --> infinity. We show that alpha(k,d) converges to a limit alpha(k) as d --> infinity for e ach k. The limit alpha(k) arose in earlier work by Aldous, as the asym ptotic proportionate number of vertices of degree k in the minimum-wei ght spanning tree on k vertices, when the edge weights are taken to be independent, identically distributed random variables. We give a grap hical alternative to Aldous's characterization of the alpha(k).