Self-averaging of perturbation Hamiltonian density in perturbed spin systems

TL;DR

The paper proves that in large disordered spin systems, the variance of the perturbation Hamiltonian density diminishes, indicating the absence of non-spontaneous replica symmetry-breaking and simplifying previous proofs.

Contribution

It provides a simpler proof that the variance of the perturbation Hamiltonian density vanishes in the infinite-volume limit under fewer assumptions.

Findings

Variance of perturbation Hamiltonian density vanishes as volume grows

Impossibility of non-spontaneous replica symmetry-breaking in these systems

Variance of spin overlap vanishes in the replica symmetric state

Abstract

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 in a simpler way and under less assumptions than before. 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.