  In this article, following a first work of the author, we study critical
subsolutions in discrete weak KAM theory. In particular, we establish that if
the cost function $c:M \times M\to \R{}$ defined on a smooth connected manifold
is locally semi-concave and verifies twist conditions, then there exists a
$C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the
Aubry set). We also explain how this applies to costs coming from Tonelli
Lagrangians. Finally, following ideas introduced in the work of Fathi-Maderna
and Mather, we study invariant cost functions and apply this study to certain
covering spaces, introducing a discrete analogue of Mather's $\alpha$ function
on the cohomology.
