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Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field by Garban, Christophe; Sepúlveda, Avelio is a scholarly article available to read on EtoBox.

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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-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

Author

Garban, Christophe; Sepúlveda, Avelio

Published

2020

Language

EN