  For an inverse temperature $\beta>0$, we define the $\beta$-circular Riesz
gas on $\mathbb{R}^d$ as any microscopic thermodynamic limit of Gibbs particle
systems on the torus interacting via the Riesz potential $g(x) = \Vert x
\Vert^{-s}$. We focus on the non integrable case $d-1<s<d$. Our main result
ensures, for any dimension $d\ge 1$ and inverse temperature $\beta>0$, the
existence of a $\beta$-circular Riesz gas which is not number-rigid. Recall
that a point process is said number rigid if the number of points in a bounded
Borel set $\Delta$ is a function of the point configuration outside $\Delta$.
It is the first time that the non number-rigidity is proved for a Gibbs point
process interacting via a non integrable potential. We follow a statistical
physics approach based on the canonical DLR equations. It is inspired by
Dereudre-Hardy-Lebl\'e and Ma\"ida (2021) where the authors prove the
number-rigidity of the $\text{Sine}_\beta$ process.

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