Asymptotics for weighted minimal spanning trees on random points

For all p[greater-or-equal, slanted]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-Xjp. If the Xi, i[greater-or-equal, slanted]1, are i.i.d. with values in [0,1]d, d[greater-or-equal, slanted]2, and have a density f which is bounded away from zero and which has support [0,1]d, then for all p[greater-or-equal, slanted]1, including p in the critical range p[greater-or-equal, slanted]d, we haveHere 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.