  Dense non-Brownian suspension flows of hard particles display mystifying
properties: as the jamming threshold is approached, the viscosity diverges, as
well as a length scale that can be identified from velocity correlations. To
unravel the microscopic mechanism governing dissipation and its connection to
the observed long-range correlations, we develop an analogy between suspension
flows and the rigidity transition occurring when floppy networks are pulled --
a transition believed to be associated to the stress-stiffening of certain
gels. After deriving the critical properties near the rigidity transition, we
show numerically that suspensions flows lie close to it. We find that this
proximity causes a decoupling between viscosity and the correlation length of
velocities \xi, which scales as the length l_c characterizing the response of
the velocity in flow to a local perturbation, previously predicted to follow
l_c\sim 1/\sqrt{z_c-z}\sim p^{0.18} where p is the dimensionless particle
pressure, z the coordination of the contact network made by the particles and
z_c is twice the spatial dimension. We confirm these predictions numerically,
predict the existence of a larger length scale l_r\sim 1/\sqrt{p} with mild
effects on velocity correlation and the existence of a vanishing strain \delta
\gamma\sim 1/p that characterizes de-correlation in flow.
