Consider lattice models in classical statistical mechanics, like the Ising model, specified by the Gibbs ensemble of a (real-valued) local lattice Hamiltonian. What's the slowest that correlation functions can decay with respect to distance?
Note the 2D Ising model at low temperature has truly infinite-range (non-decaying) correlations, at least if you consider a mixture of the two symmetry-broken Gibbs states. I want to avoid cases like this by restricting attention to states where correlations tend to zero at large distance. E.g. in the case of the Ising model, I would want to restrict attention to one of the symmetry-broken states, rather than considering their mixture. (According to Velenik, the states with decaying correlations are precisely the extremal Gibbs states, i.e. the ones that are not mixtures.) I'm then asking about how slowly this decay can occur.
In 1D, at nonzero temperature, correlation functions must decay at least exponentially. E.g. in the 1D Ising model we have a spin correlation function $\langle \sigma_x \sigma_{x+r}\rangle \propto e^{-r/\xi}$. By considering the transfer matrix, you see such decay is necessary in 1D models more generally.
In 2D, correlation functions can decay more slowly. E.g. in the 2D classical XY model, at low temperatures, correlation functions can decay with an arbitrarily slow power law (see here). Is it possible for correlations to decay more slowly than a power law in 2D?
In dimension d>2, I'm not sure of any bounds. If the lattice model were well-approximated by a Euclidean CFT at long distances, reflection positivity would imply (via "unitarity bounds" in CFT) the correlation function $\langle \phi(0)\phi(x)\rangle \propto |x|^{-2\Delta}$ satisfies $\Delta \geq (d-2)/2$, i.e. the slowest decay is a power law with power $(d-2)$. But without the CFT assumption, I'm not sure that's the slowest possible decay. Is there an argument directly from the lattice, perhaps using reflection positivity on the lattice? (I'm most interested in the case of real lattice Hamiltonians, not complex ones.)
Motivation: I'd like to more carefully understand why critical lattice models exhibit power-law correlations and scaling limits described by field theory. I think one can argue that at a phase transition, in order to exhibit non-analyticity, there cannot be a finite correlation length (similar to how phase transitions cannot occur in finite volume), i.e. correlations cannot decay exponentially. But I'm not sure how to argue for power-law correlations in particular, except by appealing to intuition based on RG. So the above question seemed like a first step toward more careful understanding.
