A heteropolymer near a linear interface

We consider a quenched-disordered heteropolymer, consisting of hydrophobic and hydrophylic monomers, in the vicinity of an oil-water interface. The he teropolymer is modeled by a directed simple random walk (i, S-i)(i is an el ement of N) on N x Z with an interaction given by the Hamiltonians H-n(omeg a)(S) = lambda Sigma(i=1)(n)(omega(i) + h)Sign(S-i) (n is an element of N). Here, lambda and h are parameters and (omega(i))(i is an element of N) are i.i.d. ii-valued random variables. The sign(S-i)= +/-1 indicates whether t he ith monomer is above or below the interface, the omega(i) = +/-1 indicat es whether the ith monomer is hydrophobic or hydrophylic. It was shown by B olthausen and den Hollander that the free energy exhibits a localization-de localization phase transition at a curve in the (lambda, h)-plane. In the p resent paper we show that the free-energy localization concept is equivalen t to pathwise localization. In particular, we prove that free-energy locali zation implies exponential tightness of the polymer excursions away from th e interface, strictly positive density of intersections with the interface and convergence of ergodic averages along the polymer. We include an argume nt due to G. Giacomin, showing that free-energy delocalization implies that there is pathwise delocalization in a certain weak sense.