  This paper is the first of a series where we study quantum channels from the
random matrix point of view. We develop a graphical tool that allows us to
compute the expected moments of the output of a random quantum channel. As an
application, we study variations of random matrix models introduced by Hayden
\cite{hayden}, and show that their eigenvalues converge almost surely. In
particular we obtain for some models sharp improvements on the value of the
largest eigenvalue, and this is shown in a further work to have new
applications to minimal output entropy inequalities.
