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.