Experimental realizations of a one-dimensional (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 $T_c\left(\xi \right)$. 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 $\overline{F}(t,y)$, which encodes the microscopic disorder integrated by the DP up to a growing time $t$ and an endpoint position $y$, its derivative $\eta (t,y)$, and their respective two-point correlators $\overline{C}(t,y)$ and $\overline{R}(t,y)$. We compute the exact solution of its linearized evolution ${\overline{R}}^{\mathrm{lin}}(t,y)$ and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder ($\xi =0$) to justify the construction of a “toy model” leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-type free-energy fluctuations, correlated at small $\left|y\right|\lesssim \xi$, and of amplitude ${\tilde{D}}_{\infty }(T,\xi )$. We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts ${\tilde{D}}_{\infty }\sim 1/T$ at high temperatures and ${\tilde{D}}_{\infty }\sim 1/T_c\left(\xi \right)$ at low temperatures. We identify the connection between the temperature-induced crossover of ${\tilde{D}}_{\infty }(T,\xi )$ and the full replica symmetry breaking in previous Gaussian variational method (GVM) computations. In order to refine our toy model with respect to finite-time geometrical fluctuations, we propose an effective time-dependent amplitude ${\tilde{D}}_t$. Finally, we discuss the consequences of the low-temperature regime for two experimental realizations of Kardar-Parisi-Zhang interfaces, namely, the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.

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