Copolymer with pinning: variational characterization of the phase diagram

TL;DR

This paper provides a detailed variational analysis of a polymer model near an interface, characterizing phase transitions between localized and delocalized states with new formulas and insights.

Contribution

It introduces variational formulas for the quenched and annealed free energies, resolving open questions about phase diagrams and polymer behavior.

Findings

Derived variational formulas for free energies

Identified critical curves for phase transitions

Characterized polymer behavior in each phase

Abstract

This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model each undergo a transition from a localized phase (where the polymer stays close to the interface) to a delocalized phase (where the polymer wanders away from the interface). We exploit the approach developed in [5] and [3] to derive variational formulas for the quenched and the annealed free energy per monomer. These variational formulas are analyzed to obtain detailed information on the critical curves separating the two phases and on the typical behavior of the polymer in each of the two phases. Our main results settle a number of open questions.