Griffiths Inequalities for Ising Spin Glasses on the Nishimori Line

The Griffiths inequalities for Ising spin glasses are proved on the Nishimori line with various bond randomness which includes Gaussian and $\pm J$ bond randomness. The proof for Ising systems with Gaussian bond randomness has already been carried out by Morita et al, which uses not only the gauge theory but also the properties of the Gaussian distribution, so that it cannot be directly applied to the systems with other bond randomness. The present proof essentially uses only the gauge theory, so that it does not depend on the detail properties of the probability distribution of random interactions. Thus, the results obtained from the inequalities for Ising systems with Gaussian bond randomness do also hold for those with various bond randomness, especially with $\pm J$ bond randomness.