  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 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.
