  Let $ \nu $ be a probability distribution over the linear semi-group $
\mathrm{End}(E) $ for $ E $ a finite dimensional vector space over a locally
compact field. We assume that $ \nu $ is proximal, strongly irreducible and
that $ \nu^{*n}\{0\}=0 $ for all integers $ n\in\mathbb{N} $. We consider the
random sequence $ \overline\gamma_n := \gamma_0 \cdots \gamma_{n-1} $ for $
(\gamma_k)_{k \ge 0} $ independents of distribution law $ \nu $. We define the
logarithmic singular gap as $ \mathrm{sqz} = \log\left( \frac{\mu_1}{\mu_2}
\right) $ , where $ \mu_1 $ and $ \mu_2 $ are the two largest singular values.
We show that $ (\mathrm{sqz}(\overline{\gamma}_n))_{n\in\mathbb{N}} $ escapes
to infinity linearly and satisfies exponential large deviations estimates below
its escape rate. Using this escape speed, we also show that the image of a
generic line by $ \overline{\gamma}_n $ as well as its eigenspace of maximal
eigenvalue both converge to the same random line $ l_\infty $ at an exponential
speed.
  If we moreover assume that the push-forward distribution $ N(\nu) $ is $
\mathrm{L}^p $ for $ N:g\mapsto\log\left(\|g\|\|g^{-1}\|\right) $ and for some
$ p\ge 1 $ , then we show that $ \log|w(l_\infty)| $ is $ \mathrm{L}^p $ for
all unitary linear form $ w $ and the logarithm of each coefficient of $
\overline{\gamma}_n $ is almost surely equivalent to the logarithm of the norm.
  To prove these results, we do not rely on any classical results for random
products of invertible matrices with $ \mathrm{L}^1 $ moment assumption.
Instead we describe an effective way to group the i.i.d factors into i.i.d
random words that are aligned in the Cartan projection. We moreover have an
explicit control over the moments.
