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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-valuedGaussian free field. In the case of the $2d$ Villain model, we prove that thefluctuations induced by the vortices are at least of the same order ofmagnitude as the ones produced by the spin-wave. We obtain the followingquantitative upper-bound on the two-point correlation in $\mathbb{Z}^2$ when$\beta>1$ \[ \langle\sigma_x \sigma_y\rangle_{\beta}^{Villain} \leq C \, \left(\frac 1 {\|x-y\|_2}\right)^{\frac 1 {2\pi \beta}\left ( 1+\betae^{-\frac{(2\pi)^2}{2} \beta}\right )} \] The proof is entirelynon-perturbative. Furthermore it provides a new and algorithmically efficientway of sampling the $2d$ Coulomb gas. For the $2d$ Coulomb gas, we obtain thefollowing 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 withthe predictions based on a RG analysis from [JKKN77] and suggests that theCoulomb potential $\Delta^{-1}q$ at inverse temperature $\beta$ should scalelike a Gaussian free field of inverse temperature of order $\exp(\pi^2 \beta)$.Finally, we transfer the above vortex fluctuations via a duality identity tothe integer-valued GFF by showing that its maximum deviates in a quantitativeway from the maximum of a usual GFF. More precisely, we show that with highprobability 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 $[-n,n]^2$ atinverse temperature $\beta^{-1}$. Applications to the free-energies of theCoulomb gas, the Villain model and the integer-valued GFF are also considered.

Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field