Polymer pinning at an interface

We consider a (1+1)-dimensional hydrophobic homopolymer, in interaction with an oil–water interface. In Z2Z2, the interface is modelled by the xx axis, the oil is above, the water is below, and the polymer configurations are given by a simple random walk (Si)i≥0(Si)i≥0. The hydrophobicity of each monomer tends to delocalize the polymer in the upper half-plane, through a reward h>0h>0 for each monomer in the oil and a penalty −h<0−h<0 for each monomer in the water. On the other hand, the chain receives a random reward (or penalty) when crossing the interface, depending on a local random charge attached to the interface. At site ii this reward is β(1+sζi)β(1+sζi), where (ζi)i≥1(ζi)i≥1 is a sequence of i.i.d. centered random variables, and s≥0s≥0, β≥0β≥0. Since the reward is positive on average, the interface attracts the polymer and a localization effect may arise. We transform the measure of each trajectory with the hamiltonian β∑i=1N(1+sζi)1{Si=0}+h∑i=1Nsign(Si), and study the critical curve hcs(β) that separates the (β,h)(β,h)-plane into a localized and a delocalized phase for ss fixed.It is not difficult to show that hcs(β)≥hc0(β) for all s≥0s≥0 with the former explicitly computable. In this article we give a method for improving in a quantitative way this lower bound. To that end, we transform the strategy developed by Bolthausen and den Hollander in [E. Bolthausen, F. den Hollander, Localization for a polymer near an interface, Ann. Probab. 25 (3) (1997) 1334–1366], by taking into account the fact that the chain can target the sites where it comes back to the origin. The improved lower bound is interesting even for the case where only the interaction at the interface is active, i.e., for the pure pinning model. Our bound improves an earlier bound of Alexander and Sidoravicius in [K. Alexander, V. Sidoravicius Pinning of polymers and interfaces by random potential, 2005 (preprint). Available on: arXiv.org e-print archive: math.PR/0501028].