Upper bounds on cluster distribution functions and the nature of the Griffiths singularity

An upper bound is obtained on the number of $N$-site clusters. This bound enables to show that the free energy of a quenched random classical system of spins interacting via short-range interactions is differentiable to all orders in the magnetic field below some finite occupation probability. It follows that at low occupation probabilities, the Griffiths singularity is an essential singularity.