Random subgraphs in Cartesian powers of regular graphs (Q426814)

Summary: Let \(G\) be a connected \(d\)-regular graph with \(k\) vertices. We investigate the behaviour of a spanning random subgraph \(G^n_p\) of \(G^n\), the \(n\)-th Cartesian power of \(G\), which is constructed by deleting each edge independently with probability \(1-p\). We prove that \(\lim\limits_{n \rightarrow \infty} \mathbb{P}[G^n_p \text{\;is \;connected}]=e^{-\lambda}\), if \(p=p(n)=1-\left(\frac{\lambda_n^{1/n}}{k}\right)^{1/d}\) and \(\lambda_n \rightarrow \lambda>0\) as \(n \rightarrow \infty\). This extends a result of \textit{L. Clark} [``Random subgraphs of certain graph powers,'' Int. J. Math. Math. Sci. 32, No. 5, 285--292 (2002; Zbl 1004.05054)].