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

Abstract

In this paper, we study the influence of the vortices on the fluctuations of systems such as the Coulomb gas, the Villain model, or the integer‐valued Gaussian free field (GFF). In the case of the 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 when The proof is entirely nonperturbative. Furthermore, it provides a new and algorithmically efficient way of sampling the Coulomb gas. For the Coulomb gas, we obtain the following lower bound on its fluctuations at high inverse temperature: This estimate coincides with the predictions based on Renormalization group (RG) analysis by José et al. [Phys. Rev. B 16 (1977), no. 3, 1217] and suggests that the Coulomb potential at inverse temperature should scale like a GFF of inverse temperature of order .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 where is an integer‐valued GFF in the box at inverse temperature . Applications to the free energies of the Coulomb gas, the Villain model, and the integer‐valued GFF are also considered.