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

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.