  In this paper, we study the influence of the vortices on the fluctuations of
$2d$ systems such as the Coulomb gas, the Villain model or the integer-valued
Gaussian free field. In the case of the $2d$ Villain model, we prove that the
fluctuations induced by the vortices are at least of the same order of
magnitude as the ones produced by the spin-wave. We obtain the following
quantitative upper-bound on the two-point correlation in $\mathbb{Z}^2$ when
$\beta>1$ \[ \langle\sigma_x \sigma_y\rangle_{\beta}^{Villain} \leq C \, \left(
\frac 1 {\|x-y\|_2}\right)^{\frac 1 {2\pi \beta}\left ( 1+\beta
e^{-\frac{(2\pi)^2}{2} \beta}\right )} \] The proof is entirely
non-perturbative. Furthermore it provides a new and algorithmically efficient
way of sampling the $2d$ Coulomb gas. For the $2d$ Coulomb gas, we obtain the
following lower bound on its fluctuations at high inverse temperature \[
\mathbb{E}_\beta^{Coul}[\langle \Delta^{-1}q, g\rangle] \geq \exp(-\pi^2 \beta
+ o(\beta)) \langle g,(-\Delta)^{-1}g \rangle \] This estimate coincides with
the predictions based on a RG analysis from [JKKN77] and suggests that the
Coulomb potential $\Delta^{-1}q$ at inverse temperature $\beta$ should scale
like a Gaussian free field of inverse temperature of order $\exp(\pi^2 \beta)$.
Finally, we transfer the above vortex fluctuations via a duality identity to
the integer-valued GFF by showing that its maximum deviates in a quantitative
way from the maximum of a usual GFF. More precisely, we show that with high
probability when $\beta>1$ \[ \max_{x\in [-n,n]^2} \Psi_n(x) \leq
\sqrt{\frac{2\beta}{\pi} \big(1 - \beta e^{- \frac{(2\pi)^2\beta} {2} } \big)}
\log n \,. \] where $\Psi_n$ is an integer-valued GFF in the box $[-n,n]^2$ at
inverse temperature $\beta^{-1}$. Applications to the free-energies of the
Coulomb gas, the Villain model and the integer-valued GFF are also considered.
