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We consider disordered pinning models, when the return time distribution ofthe underlying renewal process has a polynomial tail with exponent $\alpha \in(1/2,1)$. This corresponds to a regime where disorder is known to be relevant,i.e. to change the critical exponent of the localization transition and toinduce a non-trivial shift of the critical point. We show that the free energyand critical curve have an explicit universal asymptotic behavior in the weakcoupling regime, depending only on the tail of the return time distribution andnot on finer details of the models. This is obtained comparing the partitionfunctions with corresponding continuum quantities, through coarse-grainingtechniques.

Universality for the pinning model in the weak coupling regime