## Electronic Communications in Probability

### Connectivity threshold for random subgraphs of the Hamming graph

#### Abstract

We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $np- \log n \to - \infty$ the probability that the graph is connected tends to $0$, while when $np- \log n \to + \infty$ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $np- \log n \to t \in \mathbb{R}$. We find that the threshold does not depend on $d$, unlike the phase transition of the giant connected component of the Hamming graph (see [1]). Within the critical window, the connectivity probability does depend on $d$. We determine how.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 27, 8 pp.

Dates
Accepted: 23 February 2016
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.ecp/1457978024

Digital Object Identifier
doi:10.1214/16-ECP4479

Mathematical Reviews number (MathSciNet)
MR3485396

Zentralblatt MATH identifier
1336.05074

#### Citation

Federico, Lorenzo; van der Hofstad, Remco; Hulshof, Tim. Connectivity threshold for random subgraphs of the Hamming graph. Electron. Commun. Probab. 21 (2016), paper no. 27, 8 pp. doi:10.1214/16-ECP4479. https://projecteuclid.org/euclid.ecp/1457978024

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