This paper concerns random bipartite planar maps which are defined byassigning weights to their faces. The paper presents a threefold contributionto the theory. Firstly, we prove the existence of the local limit for allchoices of weights and describe it in terms of an infinite mobile. Secondly, weshow 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 oneface of infinite degree and has in that case spectral dimension $4/3$ (thelatter requires a mild moment condition).
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