Asymptotics for weighted minimal spanning trees on random points

For all p⩾1 let Mp(X1,…,Xn) denote the length of the minimal spanning tree through random variables X1,…,Xn, where the cost of an edge (Xi, Xj) is given by ||Xi−Xj||p. If the Xi, i⩾1, are i.i.d. with values in [0,1]d, d⩾2, and have a density f which is bounded away from zero and which has support [0,1]d, then for all p⩾1, including p in the critical range p⩾d, we havelimn→∞Mp(X1,…, Xn)/n(d−p)/d=C(p,d)∫[0,1]df(x)(d−p)/d dx c.c.Here C(p,d) denotes a positive constant depending only on p and d and c.c. denotes complete convergence. Extensions to related optimization problems are indicated and rates of convergence are also found.