## Joint distribution of distances in large random regular networks.(English)Zbl 1277.60021

Summary: We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime in which the degree is kept fixed while the number of vertices tends to $$\infty$$. The marginal distribution of an individual entry is now well understood, thanks to the work of S. Bhamidi et al. [Ann. Appl. Probab. 20, No. 5, 1907–1965 (2010; Zbl 1213.60157)]. The purpose of this note is to show that the whole array, suitably re-centered, 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.

### MSC:

 60C05 Combinatorial probability 05C80 Random graphs (graph-theoretic aspects) 90B15 Stochastic network models in operations research

Zbl 1213.60157
Full Text:

### References:

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