In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the $p$-Rényi entropy. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten $S_1 \to S_p$ norms.