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Random bond Ising systems on a general hierarchical lattice are considered.The inequality between the specific heat exponent of the pure system,$\alpha_p$, and the crossover exponent $\phi$, $\alpha_p<=\phi$, gives rise toa possibility of a negative $\alpha_p$ along with a positive $\phi$, leading torandom criticality in disagreement with the Harris criterion. An explicitexample where this really happens for an Ising system is presented anddiscussed. In addition to that, it is shown that in presence of full long-rangecorrelations, the crossover exponent is larger than in the uncorrelated case.Comment: 8 pages, 3 figure

Harris criterion on hierarchical lattices: Rigorous inequalities and counterexamples in Ising systems