The Statistical Mechanics of Stretched Polymers

TL;DR

This paper investigates the statistical properties of stretched self-interacting polymers, providing detailed phase transition analysis, local limit theorems, and recent progress on phase transition order and polymer behavior in random environments.

Contribution

It offers a precise description of the stretched phase, characterizes the critical force for phase transition, and advances understanding of polymer behavior in random environments.

Findings

Characterization of the stretched phase and critical force

Determination of phase transition order in the attractive case

Proof of diffusive behavior of semi-directed polymers in high dimensions

Abstract

We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. We concentrate mainly on the cases of purely attractive or purely repulsive self-interactions, but our results are stable under suitable small perturbations of these pure cases. We provide in particular a precise description of the stretched phase (local limit theorems for the end-point and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (non-trivial, direction-dependent) critical force needed to trigger the collapsed/stretched phase transition in the attractive case. We also describe some recent progress: first, the determination of the order of the phase transition in the attractive case; second, a proof that a semi-directed polymer in quenched random environment is diffusive in dimensions 4 and higher when the temperature is high enough. In addition, we correct an incomplete argument from one of our earlier works.