What does the correlation function look like in first-order transition?

Question

I know the correlation function for critical phenomena $$G(r)\sim \frac{1}{|r|^{d-2+\eta}}$$ for $r\ll \xi$ and $$G(r)\sim e^{-|r|/\xi}$$ for $r\gg\xi$.

Answer 1

The correlation length at a first-order phase transition remains finite in general. It follows that the large-distance asymptotic behavior of the (truncated) 2-point correlation function is usually still of the form $$ G(r) \sim \frac{1}{|r|^{(d-1)/2}}e^{-|r|/\xi}. $$ (The prefactor given above is the one predicted by Ornstein-Zernike theory and generally applies, but there are exceptions.)

Since $\xi$ remains finite, and since there is no parameter with which to play to make $\xi\gg 1$, one cannot investigate what happens when $1 \ll |r| \ll \xi$. In particular, it will not be possible to reach an asymptotic regime in which the "short" distance behavior becomes universal.