Decay of correlations and thermodynamic limit for the circular Riesz gas

We investigate the thermodynamic limit of the circular long-range Riesz gas, a system of particles interacting pairwise through an inverse power kernel. We show that after rescaling, so that the typical spacing of particles is of order $1$, the microscopic point process converges as the number of points tends to infinity, to an infinite volume measure $\mathrm{Riesz}_{s,\beta}$. This convergence result is obtained by analyzing gaps correlations, which are shown to decay in power-law with exponent $2-s$. Our method is based on the analysis of the Helffer-Sj\"ostrand equation in its static form and on various discrete elliptic regularity estimates.