  We establish spectral theorems for random walks on mapping class groups of
connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In
both cases, we relate the asymptotics of the stretching factor of the
diffeomorphism/automorphism obtained at time $n$ of the random walk to the
Lyapunov exponent of the walk, which gives the typical growth rate of the
length of a curve -- or of a conjugacy class in $F_N$ -- under a random product
of diffeomorphisms/automorphisms.
  In the mapping class group case, we first observe that the drift of the
random walk in the curve complex is also equal to the linear growth rate of the
translation lengths in this complex. By using a contraction property of typical
Teichm\"uller geodesics, we then lift the above fact to the realization of the
random walk on the Teichm\"uller space. For the case of $\text{Out}(F_N)$, we
follow the same procedure with the free factor complex in place of the curve
complex, and the outer space in place of the Teichm\"uller space. A general
criterion is given for making the lifting argument possible.
