On k-connectivity for a geometric random graph

For n points uniformly randomly distributed on the unit cube in d dimensions, with d greater than or equal to 2, let rho(n) (respectively, sigma(n)) denote the minimum r at which the graph, obtained by adding an edge between each pair of points distant at most r apart, is k-connected (respectively, has minimum degree k). Then P[rho(n) = sigma(n)] --> 1 as n --> infinity. (C) 1999 John Wiley & Sons, Inc.