The talk is a report of work in progress with Charles Favre.

Let $G$ be a finitely generated group endowed with some probability measure $\mu$ and $(\rho_\lambda)$ be an

algebraic family of representations of $G$ into $SL(2,\mathbb C)$, diverging in the representation space as $\lambda\to \infty$.

Using non-Archimedean techniques, we study the asymptotics of the random product of matrices induced by $\rho_\lambda(G, \mu)$

as $\lambda\to \infty$. In particular we can describe the growth rate of the Lyapunov exponent in terms of non-Archimedean data.