On the expected value of energy in groups

We obtain explicit upper and lower bounds for the expected action energy associated with a pair $(\mathsf{A},\mathsf{\Delta })$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $(\mathsf{A},\mathsf{A})$ and $(\mathsf{A},{\mathsf{A}}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $\mathrm{\mid }{\mathsf{A}}^{\ast 2}\mathrm{\mid }$ and $\mathrm{\mid }\mathsf{A}{\mathsf{A}}^{-1}\mathrm{\mid }$.