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 2.. systems such as the Coulomb gas, the Villain model, or the integer-valued Gaussian free field (GFF). In the case of the 2.. 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 Z(2) when. beta > 1(sic)Villain(sic) . The proof is entirely nonperturbative. Furthermore, it provides a new and algorithmically efficient way of sampling the 2.. Coulomb gas. For the 2.. Coulomb gas, we obtain the following lower bound on its fluctuations at high inverse temperature(sic) exp(sic)(sic).. This estimate coincides with the predictions based on Renormalization group (RG) analysis by Jose et al. [Phys. Rev. B 16 (1977), no. 3, 1217] and suggests that the Coulomb potential Delta(-1)q. at inverse temperature.beta