## Journal of Applied Probability

### Joint distribution of distances in large random regular networks

Justin Salez

#### 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.

#### Article information

Source
J. Appl. Probab., Volume 50, Number 3 (2013), 861-870.

Dates
First available in Project Euclid: 5 September 2013

https://projecteuclid.org/euclid.jap/1378401241

Digital Object Identifier
doi:10.1239/jap/1378401241

Mathematical Reviews number (MathSciNet)
MR3102519

Zentralblatt MATH identifier
1277.60021

Subjects
Primary: 60C05: Combinatorial probability

#### Citation

Salez, Justin. Joint distribution of distances in large random regular networks. J. Appl. Probab. 50 (2013), no. 3, 861--870. doi:10.1239/jap/1378401241. https://projecteuclid.org/euclid.jap/1378401241

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