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 ∞. 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 an explicit 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 along the edges.

Great variations were observed in length-at-age among populations of white sucker, Catostomus commersoni, in 32 Ontario lakes. The spatial structuring of these data was examined using the Mantel test with respect to the corresponding spatial structuring for various environmental variables which might affect growth of the white sucker. These environmental variables include lake geographic location, lake morphometry, water chemistry, food supply, presence/absence of major predator species, and population density and length at sexual maturity of the white sucker. Geographic distances among lakes and among-lake differences in water chemistry were significantly related to among-population differences in length-at-age. Among-lake differences in lake morphometric variables, benthos densities, presence/absence of predator species, and length at maturity and population density of the white sucker were not significantly related to among-population differences in length-at-age. No sex-specific differences in the effects of environmental variables on length-at-age were observed. This study suggests that the among-lake differences in water chemistry (thus, physiological stresses) and isolation-by-distance (thus, genetic forces) are the two most important factors in patterning the large variations in length-at-age among white sucker populations. However, discerning the separate effect of each of these two factors is not possible because the spatial patterns of these two factors are related.