We consider a periodic version of the Riesz gas consisting of N classical particles on a circle, interacting via a two-body repulsive potential which behaves locally as a power law of the distance, \(\sim g/|x|^s\) for \(s>-1\) . Long range (LR) interactions correspond to \(s<1\) , short range (SR) interactions to \(s>1\) , while the cases \(s=0\) and \(s=2\) describe the well-known log-gas and the Calogero–Moser (CM) model respectively. We study the fluctuations of the positions around the equally spaced crystal configuration, both for Brownian particles—passive noise—and for run-and-tumble particles (RTP)—active noise. We focus on the weak noise regime where the equations of motion can be linearized, and the fluctuations can be computed using the Hessian matrix. We obtain exact expressions for the space-time correlations, both at the macroscopic and microscopic scale, for \(N \gg 1\) and at fixed mean density \(\rho \) . They are characterized by a dynamical exponent \(z_s=\min (1+s,2)\) . We also obtain the gap statistics, described by a roughness exponent \(\zeta _s=\frac{1}{2} \min (s,1)\) . For \(s>0\) in the Brownian case, we find that in a broad window of time, i.e. for \(\tau =1/(g \rho ^{s+2}) \ll t \ll N^{z_s} \tau \) , the root mean square displacement of a particle exhibits sub-diffusion as \(t^{1/4}\) for SR as in single-file diffusion, and \(t^{\frac{s}{2(1+s)}}\) for LR interactions. Remarkably, this coincides, including the amplitude, with a recent prediction obtained using macroscopic fluctuation theory. These results also apply to RTPs beyond a characteristic time-scale \(1/\gamma \) , with \(\gamma \) the tumbling rate, and a length-scale \({\hat{g}}^{1/z_s}/\rho \) with \({\hat{g}}=1/(2\gamma \tau )\) . Instead, for either shorter times or shorter distances, the active noise leads to a rich variety of static and dynamical regimes, with distinct exponents, for which we obtain detailed analytical results. For \(-1<s<0\) , the displacements are bounded, leading to true crystalline order at weak noise. The melting transition, recently observed numerically, is discussed in light of our calculation. Finally, we extend our method to the active Dyson Brownian motion and to the active Calogero–Moser model in a harmonic trap, generalizing to finite \(\gamma \) the results of our earlier work. Our results are compared with the mathematics literature whenever possible.
