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Pinning models are built from discrete renewal sequences by rewarding (orpenalizing) the trajectories according to their number of renewal epochs up totime $N$, and $N$ is then sent to infinity. They are statistical mechanicsmodels to which a lot of attention has been paid both because they are veryrelevant for applications and because of their {\sl exactly solvablecharacter}, while displaying a non-trivial phase transition (in fact, alocalization transition). The order of the transition depends on the tail ofthe inter-arrival law of the underlying renewal and the transition iscontinuous when such a tail is sufficiently heavy: this is the case on which wewill focus. The main purpose of this work is to give a mathematical treatmentof the {\sl finite size scaling limit} of pinning models, namely studying thelimit (in law) of the process close to criticality when the system size isproportional to the correlation length.

Finite size scaling for homogeneous pinning models