  We consider sequences of finite weighted random graphs that converge locally
to unimodular i.i.d. weighted random trees. When the weights are atomless, we
prove that the matchings of maximal weight converge locally to a matching on
the limiting tree. For this purpose, we introduce and study unimodular
matchings on weighted unimodular random trees as well as a notion of optimality
for these objects. In this context, we prove that, in law, there is a unique
optimal unimodular matching for a given unimodular tree. We then prove that
this law is the local limit of the sequence of matchings of maximal weight.
Along the way, we also show that this law is characterised by an equation
derived from a message passing algorithm.
