Extremes for the minimal spanning tree on normally distributed points

Let n points be placed independently in v-dimensional space according to the standard v-dimensional normal distribution. Let M(n) be the longest edge-length of the minimal spanning tree on these points; equivalently let M(n) be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)(1/2)M(n) - b(n) converges weakly to the Gumbel (double exponential) distribution, where b(n) are explicit constants with b(n) similar to (v - 1) log log n. We also show the same result holds if M(n) is the longest edge-length for the nearest neighbour graph on the points.