Long paths in first passage percolation on the complete graph I. Local PWIT dynamics · arXivDesk
1512.06152Dec 18, 2015This is the first paper in a series of two papers on "Long paths in first passage percolation on the complete graph". We have update the archive reference to the second paper
Long paths in first passage percolation on the complete graph I. Local PWIT dynamics
M. Eckhoff, J. Goodman, R. van der Hofstad, F. R. Nardi
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 (sn)n≥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
Nearby in the stack
-dependent edge weights. The simplest example consists of edge weights of the form
Esn
, where
E
is an exponential random variable with mean 1. In this paper, we investigate the case where
sn→∞
, 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.