Fixed point stability and decay of correlations

In the framework of the renormalization-group (RG) theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective Phi(4) theories, having an N-component fundamental field Phi(i) and containing up to fourth-order powers of the field components. Their RG flow is usually characterized by several fixed points (FPs). We give here strong arguments in favour of the following conjecture: the stable FP corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent eta describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the epsilon-expansion. Then, we discuss its validity beyond the epsilon-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.