# Infrared bound on Ising model

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.

• That makes a lot of sense now. Thanks! And just a follow up, it seems that the proof is only for $d>2$ dimensions, but it would seem that one should be able to find something similar for $d=1,2$ dimensions. I know there is a proof for the planar XY model (McBryan and Spencer), but that is only for sufficiently low temperatures. Do you know of more general proofs? Commented Jan 5, 2023 at 0:17
• In dimension $1$, the decay is always exponential, so it is not clear to me what bound you'd like in this case. In dimension $2$, at least for the planar Ising model, one has explicit expressions for the two-point function in all regimes. Concerning the McBryan-Spencer argument, it's true that it relies on $\beta$ being large, but I would be surprised if a slight variant could not be used to get power law decay at all $\beta$; note that, at small $\beta$, the bound is again poor, since the decay is exponential in this case. Commented Jan 5, 2023 at 10:16
• For the planar Ising model at $\beta<\beta_c$, one can also directly relate the 2-point function of the model with the Green function of a (killed) random walk, using the parafermionic observable. This is done in this paper. Commented Jan 5, 2023 at 10:23
• One last comment: for finite-range interactions, there is an alternative to the McBryan-Spencer bound, due to Dobrushin and Shlosman, which does not require $\beta$ to be large. The argument is described (in a more general setting) in this paper; see Theorem 1. Commented Jan 5, 2023 at 11:28