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The 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 \(1+ \alpha \;\geqslant \;1\). 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 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.

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