Long paths in first passage percolation on the complete graph I. Local PWIT dynamics (1512.06152v2)

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 [9]. 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 investigate the case where $s_n\rightarrow \infty$, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices.