Logarithmic asymptotic of multi-level Hermite-Pad\'e polynomials

We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Pad\'e approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Pad\'e approximants.