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We investigate the thermodynamic limit of the circular long-range Riesz gas,a system of particles interacting pairwise through an inverse power kernel. Weshow 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 toinfinity, to an infinite volume measure $\mathrm{Riesz}_{s,\beta}$. Thisconvergence result is obtained by analyzing gaps correlations, which are shownto decay in power-law with exponent $2-s$. Our method is based on the analysisof the Helffer-Sj\"ostrand equation in its static form and on various discreteelliptic regularity estimates.

Decay of correlations and thermodynamic limit for the circular Riesz gas