A refinement of Simon's correlation inequality

A general formulation is given of Simon's Ising model inequality: {Mathematical expression} where B is any set of spins separating α from γ. We show that 〈σ_bσ_α〉 can be replaced by 〈σ_bσ_α〉_A where A is the spin system "inside"B containing α. An advantage of this is that a finite algorithm can be given to compute the transition temperature to any desired accuracy. The analogous inequality for plane rotors is shown to hold if a certain conjecture can be proved. This conjecture is indeed verified in the simplest case, and leads to an upper bound on the critical temperature. (The conjecture has been proved in general by Rivasseau. See notes added in proof.)