Inequalities for the local energy of random Ising models

We derive a rigorous lower bound on the average local energy for the Ising model with quenched randomness. The result is that the lower bound is given by the average local energy calculated for the isolated case, i.e. in the absence of all other interactions. The only condition for this statement to hold is that the distribution function of the random interaction corresponding to the average local energy under consideration is an even function of randomness (symmetric). All other interactions can be arbitrarily distributed including non-random cases. A non-trivial fact is that any introduction of other interactions to the isolated case always leads to an increase of the average local energy, which is opposite to ferromagnetic systems where the Griffiths inequality holds. Another inequality is proved for asymmetrically distributed interactions: When we consider the value of the thermal average of local energy in the configurational space of randomness, the probability for this value to be lower than that for the isolated case takes a maximum value on the Nishimori line as a function of the temperature. In this sense the system is most stable on the Nishimori line.