Limits of balanced metrics on vector bundles and polarised manifolds

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter \alpha, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of \alpha, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einstein metric on E and a constant scalar curvature K\"ahler metric in c_1(L). For special values of \alpha, limits of balanced metrics are solutions of a system of coupled equations relating a Hermitian-Einstein metric on E and a K\"ahler metric in c_1(L). For this, we compute the top two terms of the density of states expansion of the Bergman kernel of E \otimes L^k.