I will explain how free probability, which is a theory of independence for non-commutative random variables, can be applied to understand the spectra of various models of random matrices.

We give a general construction of triangulations, starting from a walk with small steps in the quarter plane This construction is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.