Random subgraphs of Cayley graphs over \(p\)-groups (Q1590219)

Cayley graphs \(X_n\) on minimal generating sets \(S_n\) over a class of \(p\)-groups \(G_n\), namely ones whose order is bounded by \(b^{n\log n}\) and whose quotients with respect to their Frattini subgroups are isomorphic to a vector space of dimension \(n\) over the field on \(p\) elements, are considered. Random elements of \(G_n\) are selected with independent probabilities \(\lambda_n\), and the size of the induced random subgraphs of \(X_n\) is studied. By using the expander properties of the Cayley graph and information from the homomorphism that maps \(G_n\) onto \(F_p^n\), it is shown that there exists a positive constant \(c\) such that the largest component of such a random induced subgraph of \(X_n\) contains almost all vertices of \(X_n\) if (for \(p>2\)), \(\lambda_n= c \ln |2n|/|2n|\).