Multidimensional Random Polymers: A Renewal Approach

Abstract

In these lecture notes we discuss ballistic phase of quenched and annealed stretched polymers in random environment on \(\mathbb{Z}^{d}\) with an emphasis on the natural renormalized renewal structures which appear in such models. In the ballistic regime an irreducible decomposition of typical polymers leads to an effective random walk reinterpretation of the latter. In the annealed case the Ornstein-Zernike theory based on this approach paves the way to an essentially complete control on the level of local limit results and invariance principles. In the quenched case, the renewal structure maps the model of stretched polymers into an effective model of directed polymers. As a result one is able to use techniques and ideas developed in the context of directed polymers in order to address issues like strong disorder in low dimensions and weak disorder in higher dimensions. Among the topics addressed: Thermodynamics of quenched and annealed models, multi-dimensional renewal theory (under Cramer’s condition), renormalization and effective random walk structure of annealed polymers, very weak disorder in dimensions d ≥ 4 and strong disorder in dimensions d = 2, 3.