Probing Phase Transitions of Finite Directed Polymers near a Corrugated Wall via Two-Replica Analysis

We study the pinning transition in a (1+1)-dimensional lattice model of a fluctuating interface interacting with a corrugated impenetrable wall. The interface is modeled as an N-step directed one-dimensional random walk on the half-line x≥0. Its interaction with the wall is described by a quenched, site-dependent, short-ranged random potential uj (j=1,…,N), distributed according to Q(uj) and localized at x=0. By computing the first two disorder-averaged moments of the partition function, ⟨GN⟩ and ⟨GN2⟩, and by analyzing the analytic structure of the resulting expressions, we derive an explicit criterion for the coincidence or distinction of the pinning transitions in annealed and quenched systems. We show that, although the transition points of the annealed and quenched systems are always different in the thermodynamic limit, for finite systems there exists a “gray zone” in which this difference is hardly detectable. Our results may help reconcile conflicting views on whether quenched disorder is marginally relevant.