  The generalized copolymer model is a disordered system built on a discrete
renewal process with inter-arrival distribution that decays in a regularly
varying fashion with exponent $1+ \alpha\geq 1$. It exhibits a localization
transition which can be characterized in terms of the free energy of the model:
the free energy is zero in the delocalized phase and it is positive in the
localized phase. This transition, which is observed when tuning the mean $h$ of
the disorder variable, has been tackled in the physics literature notably via a
renormalization group procedure that goes under the name of \emph{strong
disorder renormalization}. We focus on the case $\alpha=0$ -- the critical
value $h_c(\beta)$ of the parameter $h$ is exactly known (for every strength
$\beta$ of the disorder) in this case -- and we provide precise estimates on
the critical behavior. Our results confirm the strong disorder renormalization
group prediction that the transition is of infinite order, namely that when
$h\searrow h_c(\beta)$ the free energy vanishes faster than any power of
$h-h_c(\beta)$. But we show that the free energy vanishes much faster than the
physicists' prediction.
