In this paper, we study the complete graph Kn with n vertices, where we
attach an independent and identically distributed (i.i.d.) weight to each of
the n(n-1)/2 edges. We focus on the weight Wn and the number of edges Hn
of the minimal weight path between vertex 1 and vertex n. It is shown in (Ann.
Appl. Probab. 22 (2012) 29-69) that when the weights on the edges are i.i.d.
with distribution equal to that of Es, where s>0 is some parameter, and E
has an exponential distribution with mean 1, then Hn is asymptotically
normal with asymptotic mean slogn and asymptotic variance s2logn. In
this paper, we analyze the situation when the weights have distribution
E−s,s>0, in which case the behavior of Hn is markedly different as
Hn is a tight sequence of random variables. More precisely, we use the
method of Stein-Chen for Poisson approximations to show that, for almost all
s>0, the hopcount Hn converges in probability to the nearest integer of
s+1 greater than or equal to 2, and identify the limiting distribution of the
recentered and rescaled minimal weight. For a countable set of special s values
denoted by S={sj}j≥2, the hopcount Hn takes on the
values j and j+1 each with positive probability.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ402 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm