  A fundamental problem for wireless ad hoc networks is the assignment of
suitable transmission powers to the wireless devices such that the resulting
communication graph is connected. The goal is to minimize the total transmit
power in order to maximize the life-time of the network. Our aim is a
probabilistic analysis of this power assignment problem. We prove complete
convergence for arbitrary combinations of the dimension d and the
distance-power gradient p. Furthermore, we prove that the expected
approximation ratio of the simple spanning tree heuristic is strictly less than
its worst-case ratio of 2.
  Our main technical novelties are two-fold: First, we find a way to deal with
the unbounded degree that the communication network induced by the optimal
power assignment can have. Minimum spanning trees and traveling salesman tours,
for which strong concentration results are known in Euclidean space, have
bounded degree, which is heavily exploited in their analysis. Second, we apply
a recent generalization of Azuma-Hoeffding's inequality to prove complete
convergence for the case p>=d for both power assignments and minimum spanning
trees (MSTs). As far as we are aware, complete convergence for $p > d$ has not
been proved yet for any Euclidean functional.
