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We establish relations between different characterizations of order in spinglass models. We first prove that the broadening of the replica overlapdistribution indicated by a nonzero standard deviation of the replica overlap$R^{1,2}$ implies the non-differentiability of the two-replica free energy withrespect to the replica coupling parameter $\lambda$. In $\mathbb Z_2$ invariantmodels such as the standard Edwards-Anderson model, the non-differentiabilityis equivalent to the spin glass order characterized by a nonzeroEdwards-Anderson order parameter. This generalization of Griffiths' theorem isproved for any short-range spin glass models with classical bounded spins. Wealso prove that the non-differentiability of the two-replica free energymentioned above implies replica symmetry breaking in the literal sense, i.e., aspontaneous breakdown of the permutation symmetry in the model with threereplicas. This is a general result that applies to a large class of random spinmodels, including long-range models such as the Sherrington-Kirkpatrick modeland the random energy model.

Griffiths-type theorems for short-range spin glass models