  We consider products of a i.i.d. sequence in a set $\{f_1,\ldots,f_m\}$ of
preserving orientation diffeomorphisms of the circle. we can naturally
associate a Lyapunov exponent $\lambda$. Under few assumptions, it is known
that $\lambda\leq 0$ and that the equality holds if and only if
$f_1,\ldots,f_m$ are simultaneously conjugated to rotations. In this paper, we
state a quantitative version of this fact in the case where $f_1,\ldots,f_m$
are $C^k$ perturbations of rotations with rotation numbers
$\rho(f_1),\ldots,\rho(f_m)$ satisfying a simultaneous diophantine condition in
the sense of Moser: we give a precise estimate on $\lambda$ (Taylor expansion)
and we prove that there exists a diffeomorphism $g$ and rotations $r_i$ such
that $\mbox{dist}(gf_ig^{-1},r_i)\ll |\lambda|^{\frac{1}{2}}$ for $i=1,\ldots
m$. We also state analog results for random products of matrices $2\times 2$,
without diophantine condition.
