Weak disorder in the stochastic mean-field model of distance II

Abstract

In this paper, we study the complete graph $K_n$ 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 $W_n$ and the number of edges $H_n$ 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 $E^s$, where $s>0$ is some parameter, and E has an exponential distribution with mean 1, then $H_n$ is asymptotically normal with asymptotic mean $s\log n$ and asymptotic variance $s^2\log n$. In this paper, we analyze the situation when the weights have distribution $E^{-s},s>0$, in which case the behavior of $H_n$ is markedly different as $H_n$ 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 $H_n$ 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 $\mathcal{S}=\{s_j\}_{j\geq2}$, the hopcount $H_n$ takes on the values j and j+1 each with positive probability.