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We consider the O(n) loop model on tetravalent maps and show how to rephraseit into a model of bipartite maps without loops. This follows from acombinatorial decomposition that consists in cutting the O(n) modelconfigurations along their loops so that each elementary piece is a map thatmay have arbitrary even face degrees. In the induced statistics, these maps aredrawn according to a Boltzmann distribution whose parameters (the face weights)are determined by a fixed point condition. In particular, we show that thedense and dilute critical points of the O(n) model correspond to bipartite mapswith large faces (i.e. whose degree distribution has a fat tail). There-expression of the fixed point condition in terms of linear integralequations allows us to explore the phase diagram of the model. In particular,we determine this phase diagram exactly for the simplest version of the modelwhere the loops are "rigid". Several generalizations of the model arediscussed.

A recursive approach to the O(n) model on random maps via nested loops