Fluctuations of one-dimensional interface in the directed polymer formulation: role of a finite interface width

An elastic interface living in a disordered medium always exhibits geometrical fluctuations, characterized in particular by the distribution of its relative displacements as a function of the lengthscale $r$, whose variance defines the interface roughness $B(r)$. Those fluctuations are the manifestation of the probability and associated effective free-energy of the different configurations of the interface, in presence of disorder and at finite temperature. Focusing specifically on the one-dimensional interface, we use the exact mapping of the static interface on the directed polymer in random medium in order to explore both analytically and numerically the role of a finite interface width $\xi>0$, assuming a short-range elasticity and a random-bond quenched disorder. Confirming the existence of a low-temperature regime where the finite microscopic width plays a crucial role, as predicted by previous Gaussian-Variational-Method predictions [Phys.Rev.B 82, 184207 (2010)], we propose a coherent picture of the physics at stake, compatible both with numerical computations and generic scaling arguments.