I'm currently trying to understand aspects of Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice. In section 4.3, he claims that for the Ising model in $\mathbb{Z}^d$, the infrared bound tells us that if $\beta <\beta_c$, then $$ \langle \sigma_x \sigma_y\rangle^\text{f} \le \frac{C}{\beta} G(x,y) $$ Where the superscript f denotes free boundary conditions, and $G(x,y)$ is the Green function of the simple random walk.
I don't quite see how this follows readily from the infrared bound. Indeed, he references Fröhlich, Simon and Spencer's paper (Comm. Math. Phys. 50(1): 79-95 (1976)), but even within that paper, the authors say "Caution. Alas (2.9) does not imply $\langle \sigma_a\sigma_b\rangle -\langle \sigma_a\rangle^2$ is pointwise bounded by $(-\Delta)^{-1}_{ab}$". Any insight would be appreciated.
Just a reminder. The infrared bound tells us that on a finite $d$-dim torus, we have $$ \langle (\sigma(-\Delta h))^2\rangle\le\frac{1}{2\beta}(-h\Delta h) $$ Take $h=e^{ikx}$ and the usual form follows. We can also take $h$ such that $-\Delta h =\delta_x -\delta_y$. Then we do indeed get some information about the correlations $\langle \sigma_x \sigma_y\rangle$. However, it gives us a lower bound, which isn't quite surprising since the infrared bound is used to prove spontaneous symmetry breaking in $d\ge 3$ dimensions.
