Dynamical fluctuations in the Riesz gas

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the $\mathrm{\mid }x-y{\mathrm{\mid }}^{-s}$ Riesz potential, causing the over-damped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for $0<s<1$, the standard deviations of both quantities grow as $t^{\frac{s}{2(1+s)}}$. When $s>1$, the interactions are effectively short-ranged, and the universal sub-diffusive $t^{\frac{1}{4}}$ growth emerges with only amplitude depending on the exponent. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.