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 → −∞ the probability that the graph is connected goes to 0, while when np − log n → +∞ it converges to 1. We also investigate the connectivity probability inside the critical window, namely when np − log n → t ∈ 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.

Expansion of Percolation Critical Points for Hamming Graphs

“…The Hamming graph is an excellent example for investigating the universality class of the ERRG, since it has a non-trivial geometry yet is highly mean-field. See [24,25,17,35] for a small sample of the literature from this perspective. A crucial motivation for the present paper is that it serves as a companion paper to [16], where we establish the scaling limit of the cluster sizes of the largest clusters within the critical window.…”

“…We use the (now) familiar path-counting estimates to bound the right-hand side of (5.16) by 17) where the factors (k + 1) in the second and third term on the right-hand side are due to interchanging the sum over k with the sum over x. The term M 1 is the contribution from walks that are constrained to remain within one line, and is bounded by…”