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Analyticity of the energy in an Ising spin glass with correlated disorder

The average energy of the Ising spin glass is known to have no singularity along a special line in the phase diagram although there exists a critical point on the line. This result on the model with uncorrelated disorder is generalized to the case with correlated disorder. For a class of correlations in disorder that suppress frustration, we show that the average energy in a subspace of the phase diagram is expressed as the expectation value of a local gauge variable of the $Z_2$ gauge Higgs model, from which we prove that the average energy has no singularity although the subspace is likely to have a phase transition on it. This is a generalization of the result for uncorrelated disorder. Though it is difficult to obtain an explicit expression of the energy in contrast to the case of uncorrelated disorder, an exact closed-form expression of a physical quantity related to the energy is derived in three dimensions using a duality relation. Identities and inequalities are proved for the specific heat and correlation functions.