Quantitative estimates for the flux of TASEP with dilute site disorder

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.