Long paths in first passage percolation on the complete graph II. Global branching dynamics

Abstract

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [6]. We describe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$ that quantifies the extreme-value behavior of small weights, and that describes different universality classes for first passage percolation on the complete graph. We consider both $n$-independent as well as $n$-dependent edge weights. The simplest example consists of edge weights of the form $E^{s_n}$, where $E$ is an exponential random variable with mean 1. In this paper, we focus on the case where $s_n\rightarrow \infty$ with $s_n=o(n^{1/3})$. Under mild regularity conditions, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices, and we prove that the number of edges in this path obeys a central limit theorem with asymptotic mean $s_n\log{(n/s_n^3)}$ and variance $s_n^2\log{(n/s_n^3)}$. This settles a conjecture of Bhamidi and van der Hofstad [6]. The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in [14]; the current article focuses on the global branching dynamics.