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

Understand

We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops.

• This follows from a combinatorial decomposition that consists in cutting the O(n) model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees.
• In the induced statistics, these maps are drawn according to a Boltzmann distribution whose parameters (the face weights) are determined by a fixed point condition.
• In particular, we show that the dense and dilute critical points of the O(n) model correspond to bipartite maps with large faces (i.e.