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We study the fluctuations of the directed polymer in 1+1 dimensions in aGaussian random environment with a finite correlation length {\xi} and atfinite temperature. We address the correspondence between the geometricaltransverse fluctuations of the directed polymer, described by its roughness,and the fluctuations of its free-energy, characterized by its two-pointcorrelator. Analytical arguments are provided in favor of a generic scaling lawbetween those quantities, at finite time, non-vanishing {\xi} and explicittemperature dependence. Numerical results are in good agreement both forsimulations on the discrete directed polymer and on a continuous directedpolymer (with short-range correlated disorder). Applications to recentexperiments on liquid crystals are discussed.

Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations