Entropy fluctuations for directed polymers in 2+1 dimensions.

We find numerically that the sample to sample fluctuation of the entropy DeltaS is a more sensitive tool in distinguishing low from high temperature behaviors than the common corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures, since (DeltaS)(2) is always extensive. In 2+1 dimensions we find a behavior that at first sight might appear to be a transition from a low temperature phase where (DeltaS)(2) is extensive to a high temperature phase where it is subextensive. This is observed in spite of the relatively large system we use. The observed behavior is explained not as a phase transition but as a strong crossover behavior. We use an analytical argument to obtain (DeltaS)(2) for high temperature, and find that while it is always extensive it is also extremely small, and that the leading extensive part decays very quickly to zero as the temperature is increased.