A Refinement of Simon’s Correlation Inequality

A general formulation is given of Simon’s Ising model inequality : $$ \left\langle {{\sigma _\alpha }{\sigma _\gamma }} \right\rangle \le \sum\limits_{b \in B} {\left\langle {{\sigma _\alpha }{\sigma _b}} \right\rangle \left\langle {{\sigma _b}{\sigma _\gamma }} \right\rangle }$$ Where B is any set of spins separating a from δ. We show that (σ{α}σb) can be replaced by (σασb)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.