A copolymer near a selective interface: variational characterization of the free energy

In this paper we consider a two-dimensional copolymer consisting of a random concatenation of hydrophobic and hydrophilic monomers near a linear interface separating oil and water acting as solvents. The configurations of the copolymer are directed paths that can move above and below the interface. The interaction Hamiltonian, which rewards matches and penalizes mismatches of the monomers and the solvents, depends on two parameters: the interaction strength $\beta\geq 0$ and the interaction bias $h \geq 0$. The quenched excess free energy per monomer $(\beta,h) \mapsto g^\mathrm{que} (\beta,h)$ has a phase transition along a quenched critical curve $\beta \mapsto h^\mathrm{que}_c(\beta)$ separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive a variational expression for $g^\mathrm{que}(\beta,h)$ by applying the quenched large deviation principle for the empirical process of words cut out from a random letter sequence according to a random renewal process. We compare this variational expression with its annealed analogue, describing the annealed excess free energy $(\beta,h) \mapsto g^\mathrm{ann}(\beta,h)$, which has a phase transition along an annealed critical curve $\beta \mapsto h^\mathrm{ann}_c(\beta)$. Our results extend to a general class of disorder distributions and directed paths. We show that $g^\mathrm{que}(\beta,h)<g^\mathrm{ann}(\beta,h)$ for all $(\beta,h)$ in the localized phase of the annealed model. We also show that $h^\mathrm{ann}_c(\beta\alpha)<h^\mathrm{que}_c (\beta)<h^\mathrm{ann}_c(\beta)$ for all $\beta>0$ when $\alpha>1$. This gap vanished when $\alpha=1$.