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

Sedimentation of random suspensions and the effect of hyperuniformity