Directed polymers at finite temperatures in 1+1 and 2+1 dimensions

We present systematic numerical simulations for directed polymers at finite temperatures in 1+1 and 2+1 dimensions. The transverse fluctuations and free energy fluctuations tend to the strong coupling limit at any temperature in both 1+1 and 2+1 dimensions for long time t. Two different definitions for energy fluctuations at finite temperatures, which are the ensemble energy fluctuations and the internal energy fluctuations, are investigated. Apart from zero temperature, the behavior of the energy fluctuations and the free energy fluctuations for directed polymers is shown to be different. At finite temperatures, the ensemble energy fluctuations in both 1+1 and 2+1 dimensions and internal energy fluctuations in 1+1 dimensions scale as t^1/2 where the free energy fluctuations in 1+1 dimensions and 2+1 dimensions scale as t^1/3 and t^0.2 respectively. As a consequence of that the specific heat in both 1+1 and 2+1 dimensions scales as t and the entropy fluctuations in 1+1 dimensions scale as t^1/2 at any finite temperature.