Connectivity Threshold for random subgraphs of the Hamming graph

TL;DR

This paper analyzes the phase transition of connectivity in random subgraphs of the Hamming graph, identifying the critical window and showing the threshold's independence from dimension, with detailed behavior within the window.

Contribution

It precisely characterizes the connectivity threshold and its independence from dimension in the Hamming graph, including the behavior within the critical window.

Findings

Connectivity threshold depends on $np - \, \log n$

Probability of connectivity transitions from 0 to 1 across the threshold

Within the critical window, connectivity probability depends on the dimension $d$

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 goes 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 the Hamming graph (see [Bor et al, 2005]). Within the critical window, the connectivity probability does depend on d. We determine how.