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We study random pinning and copolymer models, when the return distribution ofthe underlying renewal process has a polynomial tail with finite mean. Wecompute the asymptotic behavior of the critical curves of the models in theweak coupling regime, showing that it is universal. This proves a conjecture ofBolthausen, den Hollander and Opoku for copolymer models (ref. [8]), which wealso extend to pinning models.

The Critical Curve of the Random Pinning and Copolymer Models at Weak Coupling