Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions

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. With the same assumptions, 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.