Finite size scaling for homogeneous pinning models

Julien Sohier (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time $N$, and $N$ is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their {\sl exactly solvable character}, while displaying a non-trivial phase transition (in fact, a localization transition). The order of the transition depends on the tail of the inter-arrival law of the underlying renewal and the transition is continuous when such a tail is sufficiently heavy: this is the case on which we will focus. The main purpose of this work is to give a mathematical treatment of the {\sl finite size scaling limit} of pinning models, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length.

Mots clés: Renewal Theory; Pinning Models; Criticality; Subordinators; Regenerative Sets

Domaines: Probabilités [math.PR]

Julien Sohier. Finite size scaling for homogeneous pinning models. 2009. ⟨hal-00245242v2⟩