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We study the random geometry of first passage percolation on the completegraph equipped with independent and identically distributed edge weights,continuing the program initiated by Bhamidi and van der Hofstad [9]. Wedescribe 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 describesdifferent universality classes for first passage percolation on the completegraph. 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 $s_n\rightarrow \infty$, andfocus on the local neighborhood of a vertex. We establish that thesmallest-weight tree of a vertex locally converges to the invasion percolationcluster on the Poisson weighted infinite tree. In addition, we identify thescaling limit of the weight of the smallest-weight path between two uniformvertices.

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics