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We compute the generating functions of a O(n) model (loop gas model) on arandom lattice of any topology. On the disc and the cylinder, they were alreadyknown, and here we compute all the other topologies. We find that thegenerating functions (and the correlation functions of the lattice) obey thetopological recursion, as usual in matrix models, i.e they are given by thesymplectic invariants of their spectral curve.

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies