when the system has positive density is also possible (note that on the right of $(1-\alpha) N$ it is the density of empty-sites that becomes the quantity of interest). It is thus reasonable to extend the heuristic to that case. However the best strategy to produce a surplus of particle in that case becomes more involved, as the zones with positive density of particles, which are described by the hydrodynamic evolution given in Proposition \[prop:lidro\], play a role in the optimization procedure. For this reason we did not wish to bring the speculation one step further.

Organization of the paper
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In the remainder of the paper we drop the subscript $N$ in $k_N$ in order to simplify the notation. The article is organised as follows. In Section \[Sec:Prelim\], we introduce the representation through height functions and collect a few results on the invariant measure, the spectral gap and the hydrodynamic limit of the process. In Sections \[Sec:LBLB\] and \[Sec:UBLB\], we consider the large bias case and prove respectively the lower and upper bounds of Theorem \[Th:largebias\]: While the lower bound essentially follows from the hydrodynamic limit, the upper bound is more involved and is one of the main achievement 