Decay of correlations for slowly decreasing potentials

When the interaction potential of a ferromagnet decreases like $r^{-s}$, we prove that two-point correlations (i) do not decay faster than $r^{-s}$ and (ii) decay at least like $r^{-s}$ at large magnetic field and, moreover, at least like $r^{-(s-2\mathrm{}\nu )}$ where $\nu$ is the space dimension, for any nonzero magnetic field and arbitrary temperature. Extensions of the latter result to $n$-point correlations and to other systems are indicated, and a central-limit theorem, or Gaussian limit of block-spin distribution, is mentioned for slowly decreasing ferromagnetic interactions at any nonzero magnetic field.