  We prove that the flux function of the totally asymmetric simple exclusion
process (TASEP) with site disorder exhibits a flat segment for sufficiently
dilute disorder. For high dilution, we obtain an accurate description of the
flux. The result is established undera decay assumption of the maximum current
in finite boxes, which is implied in particular by a sufficiently slow power
tail assumption on the disorder distribution near its minimum. To circumvent
the absence of explicit invariant measures, we use an original renormalization
procedure and some ideas inspired by homogenization.
