Joint Distribution of Distances in Large Random Regular Networks. (30th January 2018)

• Record Type: Journal Article
• Authors: Salez, Justin
• Is Part Of: Journal of applied probability. Volume 50:Number 3(2013)
• Page Start: 861
• Page End: 870
• Publication Date: 2018-01-30
• Subjects: Random regular graph, -- distance matrix, -- first passage percolation, -- multitype Richardson process, -- configuration model, -- branching process approximation; 60C05, -- 05C80, -- 90B15; 519.2
• DOI: 10.1239/jap/1378401241
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract : 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.