seminar: Scaling limits of non nearest neighbor pinning models in (1+1) - dimension
speaker: Jean-Dominique Deuschel (TU Berlin)abstract: Starting from a random field $\phi:\N \to \R$ with non nearest neighbor interaction of the form $V(\Delta \phi)$, for a large class of potentials $V(\cdot)$, we consider the corresponding {\sl pinning model}, that is we add a delta--pinning reward for the field to touch zero. Thus the $x$--axis can be viewed as a defect line to which the field is attracted. This can be also interpreted as a model for a homogeneous polymer chain attracted to a defect line (the $x$--axis). Denoting by $\gep \ge 0$ the intensity of the pinning reward, we show that there is a phase transition at $\gepc > 0$ between a {\sl delocalized regime} $(\gep \le \gepc)$, in which the field wanders away from the defect line, and a {\sl localized regime} $(\gep > \gepc)$, in which the field sticks very close to it. Using an approach based on renewal theory, we characterize these regimes both in terms of the free energy and of the path behavior, extracting the full scaling limits of the model. In particular, in the critical regime $(\gep = \gepc)$ we show that the rescaled field converges in distribution toward the derivative of a symmetric stable Lévy process of index~$25$. joint work with Francesco Caravenna
timetable:
Wed 7 Jun, 9:30 - 10:00, Aula Mancini
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