  In this paper, we give a quantitative estimate for the sum of the first $N$
Lyapunov exponents for random perturbations of a natural class $2N$-dimensional
volume-preserving systems exhibiting strong hyperbolicity on a large but
noninvariant subset of phase space. Concrete models covered by our setting
include systems of coupled standard maps, in both `weak' and `strong' coupling
regimes.
