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

## TL;DR

This paper investigates how a fluctuating interface interacts with a corrugated wall using a lattice model and replica analysis.

## Contribution

The study introduces a two-replica analysis to distinguish pinning transitions in annealed and quenched systems.

## Key findings

- Annealed and quenched transition points differ in the thermodynamic limit.
- Finite systems show a 'gray zone' where the transition difference is negligible.
- Results clarify the relevance of quenched disorder in phase transitions.

## Abstract

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.