Multidimensional Random Polymers : A Renewal Approach

TL;DR

This paper explores the renewal structures in multidimensional random polymers, providing a unified approach to analyze ballistic phases, disorder effects, and limit behaviors in both quenched and annealed environments.

Contribution

It introduces a renewal-based framework for multidimensional polymers, linking quenched and annealed models and applying directed polymer techniques to analyze disorder regimes.

Findings

Renewal structures facilitate analysis of polymer phases.

Effective random walk models describe annealed polymers.

Results on disorder effects in various dimensions.

Abstract

In these lecture notes, which are based on the mini-course given at 2013 Prague School on Mathematical Statistical Physics, 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 effectiverandom walk reinterpretation of the latter. In the annealed casethe 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 $d\geq 4$ and strong disorder in dimensions $d=1,2$.