# Criticality and the number of paths on a lattice

In the review "Scaling, universality, and renormalization: Three pillars of modern critical phenomena" by Stanley, he makes the following claim towards the end of the paper, which is neither derived nor cited:

So then how can correlations actually propagate an infinite distance, without requiring a series of amplification stations all along the way? We can understand such ‘‘infinite-range propagation’’ as arising from the huge multiplicity of interaction paths that connect two spins if $d > 1$ (if $d = 1$, there is no multiplicity of interaction paths, and spins order only at $T = 0$). Enumeration algorithms take into account exactly the contributions of such interaction paths of length $l$ —up to a maximum length that depends on the strength of the computer used. Remarkably accurate quantitative results are obtained if this hierarchy of exact results for successive finite values of $l$ is then extrapolated to $l = \infty$. For any $T > T_c$ , the correlation between two spins along each of the interaction paths that connect them decreases exponentially with the length of the path. On the other hand, the number of such interaction paths increases exponentially, with a characteristic length that is temperature independent, depending primarily on the lattice dimension. This exponential increase is multiplied by a ‘‘gently decaying’’ power law that is negligible except for one special circumstance which we will come to...Right at the critical point, the gently decaying power-law correction factor in the number of interaction paths, previously negligible, emerges as the victor in this stand-off between the two warring exponential effects. As a result, two spins are well correlated even at arbitrarily large separation.

This is a very appealing explanation of critical phenomena, because it seems intuitively very clear, but once I try to formulate it mathematically, I flounder, as I have no references or literature to which to refer, and his language is vague.

Specifically, can anyone point me to the literature concerning the number of paths between two points having this "gently decaying" power law factor?

Likewise unclear is what he means by the correlation between two spins along a certain path, or how the correlation between two points is just the sum over all of these correlations along specific paths.