When quenched and annealed pinning transitions coincide? A directed walk near a corrugated wall in disorders of various types

Abstract: We study the pinning transition in a (1+1)-dimensional model of a fluctuating interface interacting with a corrugated impenetrable wall. The interface is described by the $N$-step directed 1D random walk on a discrete half-line $m \ge 0$, and the interaction with the wall is modeled by a quenched site-dependent short-ranged random potential $u_j$ ($j=1,...,N$) located at $m=0$, with distribution $Q(u_j)$. By computing the first two moments, $\la G_N \ra$ and $\la G_N^2 \ra$, of the partition function $G_N$ averaged over the disorder, we show that the pinning transition for $\la G_N^2 \ra$ may or may not coincide with that of $\la G_N \ra$, depending on the details of the disorder distribution $Q(u_j)$. This result reconciles opposite viewpoints on whether the pinning transition points in models with annealed and quenched disorder coincide or not.