Joint distribution of distances in random regular networks

We study the array of point-to-point distances in random regular graphs equipped with exponential edge-lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to infinity. The marginal distribution of an individual entry is now well-understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to a rather simple infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously at the same speed along the edges (competing first-passage percolation). The result applies to both the random regular multigraph produced by the configuration model and the uniform regular simple graph. AMS 2000 subject classifications: 60C05, 05C80, 90B15.