Two-temperature statistics of free energies in (1+1) directed polymers

The joint statistical properties of two free energies computed at two different temperatures in {\it the same sample} of $(1+1)$ directed polymers is studied in terms of the replica technique. The scaling dependence of the reduced free energies difference ${\cal F} = F(T_{1})/T_{1} - F(T_{2})/T_{2}$ on the two temperatures $T_{1}$ and $T_{2}$ is derived. In particular, it is shown that if the two temperatures $T_{1} \, < \, T_{2}$ are close to each other the typical value of the fluctuating part of the reduced free energies difference ${\cal F}$ is proportional to $(1 - T_{1}/T_{2})^{1/3}$. It is also shown that the left tail asymptotics of this free energy difference probability distribution function coincides with the corresponding tail of the TW distribution.