  A sequence of invertible matrices given by a small random perturbation around
a fixed diagonal partially hyperbolic matrix induces a random dynamics on the
Grassmann manifolds. Under suitable weak conditions it is known to have a
unique invariant (Furstenberg) measure. The main result gives concentration
bounds on this measure showing that with high probability the random dynamics
stays in the vicinity of stable fixed points of the unperturbed matrix, in a
regime where the strength of the random perturbation dominates the local
hyperbolicity of the diagonal matrix. As an application, bounds on sums of
Lyapunov exponents are obtained.
