Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / e / ii / Solution

The eliminated field obeys $\varphi =-c{p}^2/(2a)$, so its connected correlation function isFor the zero-mean Gaussian field of part (i), Wick theorem givesIt is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if $C_p(r)$ has exponential factor $e^{-r/{\xi }_p}$, then $C_{\varphi }(r)$ has $e^{-2r/{\xi }_p}$ and correlation length ${\xi }_p/2$, with the algebraic prefactor also squared.