Abstract
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 under a 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.
Highlights
The flux function, called current-density relation in traffic-flow physics [12], is the most fundamental object to describe the macroscopic behavior of driven lattice gases
We develop a different approach, announced in [7], based on a renormalization method to obtain a precise information on the flux function and on the flat segment at the price of additional assumptions on the disorder distribution
We focus on the case of dilute disorder which plays a key role in the physical literature [25] as a sharp transition occurs for any arbitrarily small amount of disorder
Summary
The flux function, called current-density relation in traffic-flow physics [12], is the most fundamental object to describe the macroscopic behavior of driven lattice gases. Like many shape theorems [29], our results partially extends to LPP with more general distributions Another interpretation of our renormalization scheme (see [7, Section 3.3.2]) is that it consists in a hierarchy of homogenization problems for scalar conservation laws which approximate the particle system in blocks of mesoscopic size. As explained in [7], the homogenization of a one-dimensional scalar conservation law with a “fast” flux and a “slow” flux is seen to produce a flat segment as long as the fluxes in each block are bell-shaped (but not necessarily concave) With this bell-shape assumption on the flux function, the emergence of antishocks, previously established in [37] for TASEP with a slow bond, was shown (see [4]) to hold in more general asymmetric models with single localized blockage, even in the absence of mapping on a percolation problem.
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