The statistical mechanics of stretched polymers

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 selfinteractions, 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 endpoint and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (nontrivial, 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 Ioffe and Velenik [In Analysis and Stochastics of Growth Processes and Interface Models (2008) 55–79]. © 2010, Brazilian Statistical Association. All rights reserved.