Dynamical fluctuations in the Riesz gas

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|-s Riesz potential, causing the overdamped 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 ts2(1+s). When s>1, the interactions are effectively short-ranged, and the universal subdiffusive t14 growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.