2016/09/23 by Baroni, Enrico, van der Hofstad, Remco, Komjathy, Julia
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1609.07269
We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We prove that the weight of the optimal path has tight fluctuations around the asymptotical mean of the graph-distance if and only if the following condition holds: the random variable X is such that the continuous-time branching process describing first-passage percolation exploration in the same graph with excess edge weight X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost shortest paths in the graph-distance proliferate, in the sense that there are even ones having tight total excess edge weight for various edge-weight distributions.