Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation

Experimental realizations of a 1D interface always exhibit a finite microscopic width $\xi>0$; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature $T_c(\xi)$. Exploiting the exact mapping between the static 1D interface and a 1+1 Directed Polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature $T$, with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length $\xi$. We derive the exact `time'-evolution equations of the disorder free-energy $\bar{F}(t,y)$, its derivative $\eta (t,y)$, and their respective two-point correlators $\bar{C}(t,y)$ and $\bar{R}(t,y)$. We compute the exact solution of its linearized evolution $\bar{R}^{lin}(t,y)$, and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder ($\xi=0$), to construct a `toymodel' leading to a simple description of the DP. This model is characterized by Brownian-like free-energy fluctuations, correlated at small $|y|<\xi$, of amplitude $\tilde{D}_{\infty}(T,\xi)$. We present an extended scaling analysis of the roughness predicting $\tilde{D}_{\infty} \sim 1/T$ at high-temperatures and $\tilde{D}_{\infty} \sim 1/T_c(\xi)$ at low-temperatures. We identify the connection between the temperature-induced crossover and the full replica-symmetry breaking in previous Gaussian Variational Method computations. Finally we discuss the consequences of the low-temperature regime for two experimental realizations of KPZ interfaces, namely the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.