Disorder and critical phenomena: the $\alpha=0$ copolymer model

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.