Self-averaging of perturbation Hamiltonian density in perturbed spin systems

It is shown that the variance of a perturbation Hamiltonian density vanishes in the infinite volume limit of the perturbed spin systems with quenched disorder. This is proven by a simpler way under less assumptions than the previous one. A corollary of this theorem indicates the impossibility of non-spontaneous replica symmetry breaking in disordered spin systems. The commutativity between the infinite volume limit and the switched-off limit of a replica symmetry breaking perturbation implies that the variance of the spin overlap vanishes in the replica symmetric Gibbs state.