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On the ferromagnetic and the percolative region of random spin systems

We consider d-dimensional lattice systems of bounded real-valued spins with ferromagnetic random interaction between nearest neighbours. We establish an outer and, in two dimensions, an inner bound of the parameter region where spontaneous magnetization occurs. These bounds provide an estimate of the singularity of the critical temperature at the threshold for percolation through active bonds. We derive a relationship between the ferromagnetic region and the percolative region for a correlated site–bond percolation problem, and we investigate the latter. Bounds for the level sets of the expected magnetization are also obtained.