Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.

Basically, it is a proof of the instability in an harmonic lattice crystal that uses the idea of random walk and the discrete lagrangian, and it is quite self contained. We were able to somewhat reconstruct it, but a firm reference would obviously help. If my terminology is somewhat vague, here is exactly the theorem we're trying to find its proof:

Consider the lattice $\Lambda = [-L\cdots L]^2 \in \mathbb{Z} ^2 $ and a scalar field $X$ on it, i.e. $ \varphi (x) \in \mathbb{R} $. The particles outside $\Lambda $ are tied down, meaning $ \varphi (x) = 0 $, $\forall x \notin \Lambda $.

Energy will be defined by $\nu (X) = \Sigma _{x \sim y} (\varphi (x) \ - \varphi(y))^2$, sum over all neighboring lattice points. The partition function in the regular way:

$$Z = \int\limits_{\mathbb{R} ^ {|\Lambda|} } dX \exp(-\nu (X)) $$

The theorem is as follows:

For $L \to \infty$, we have that $\langle\varphi (0)^2 \rangle = \int\limits_{\mathbb{R} ^ | \Lambda |} dX \exp(-\nu (X)) \varphi (0) ^2 $ diverges like $\log (| \Lambda | )$