Disordered pinning models and copolymers: beyond annealed bounds

TL;DR

This paper develops a new variational bound for the free energy in disordered copolymer models, demonstrating that disorder can alter critical points and confirming some physics predictions rigorously.

Contribution

It introduces a novel variational upper bound for the free energy and proves the difference between quenched and annealed critical points in certain disordered models.

Findings

Quenched critical point differs from annealed in strong disorder.

Transition from weak to strong disorder for certain pinning models.

Critical points match renormalization group predictions in specific copolymer models.

Abstract

We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9--13], pinning and wetting models in various dimensions, and the Poland--Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent $0<\alpha<1/2$ this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a "reduced wetting model" introduced by Bodineau and Giacomin [J. Statist. Phys. 117 (2004) 801--818]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.