A mixed power generating function is a one complex variable power series that is a product of powers of simpler generating functions. We introduce a geometric method to analyze the asymptotic behavior of their coefficients: for many directions $d$ in the $m$-th dimensional unit sphere we obtain asymptotics for the coefficient of $z^{n_0}$ of $\prod_{j=1}^m\{f_j(z)\}^{n_j}$ as the norm of $(n_0,n_1,...,n_m)$ tends to infinity, provided that the projection of this vector over the sphere stays sufficiently close to the direction $d$. Applications of this method to analyze the diameter of random Cayley digraphs as well as to recover known results on the core size of random planar maps will be presented during the talk.