Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, λn. Our main result is that for any minimal generating set of transpositions, for probabilities λn=1+ϵnn−1 where n−13+δ≤ϵn<1 and δ>0, a random induced subgraph has a.s. a unique largest component of size (1+o(1))⋅x(ϵn)⋅1+ϵnn−1⋅n!. Here x(ϵn) is the survival probability of a Poisson branching process with parameter λ=1+ϵn.