  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.
