  This work is concerned with the mathematical analysis of the bulk rheology of
random suspensions of rigid particles settling under gravity in viscous fluids.
Each particle generates a fluid flow that in turn acts on other particles and
hinders their settling. In an equilibrium perspective, for a given ensemble of
particle positions, we analyze both the associated mean settling speed and the
velocity fluctuations of individual particles. In the 1970s, Batchelor gave a
proper definition of the mean settling speed, a 60-year-old open problem in
physics, based on the appropriate renormalization of long-range particle
contributions. In the 1980s, a celebrated formal calculation by Caflisch and
Luke suggested that velocity fluctuations in dimension $d=3$ should diverge
with the size of the sedimentation tank, contradicting both intuition and
experimental observations. The role of long-range self-organization of
suspended particles in form of hyperuniformity was later put forward to explain
additional screening of this divergence in steady-state observations. In the
present contribution, we develop the first rigorous theory that allows to
justify all these formal calculations of the physics literature.
