Topological expansion of mixed correlations in the Hermitian 2-matrix model and x-y symmetry of the Fg algebraic invariants

We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e. a generating function for counting bicolored discrete surfaces with non-uniform boundary conditions. As an application, we prove the x-y symmetry of Eynard and Orantin (2007 Invariants of algebraic curves and topological expansion Preprint math-ph/ 0702045).