  This paper concerns random bipartite planar maps which are defined by
assigning weights to their faces. The paper presents a threefold contribution
to the theory. Firstly, we prove the existence of the local limit for all
choices of weights and describe it in terms of an infinite mobile. Secondly, we
show that the local limit is in all cases almost surely recurrent. And thirdly,
we show that for certain choices of weights the local limit has exactly one
face of infinite degree and has in that case spectral dimension $4/3$ (the
latter requires a mild moment condition).
