Entropy fluctuations for directed polymers in 2+1 dimensions
  We find numerically that the sample to sample fluctuation of the entropy, $\Delta S$, is a tool more sensitive in distinguishing how from high temperature behaviors, than the corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures since $(\Delta S)^{2}$ is always extensive. In 2+1 dimensions we find a behavior may look at first sight as a transition from a low temperature phase where $(\Delta S)^{2}$ 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 agreement to obtain $(\Delta S)^{2}$ for high temperature and find that while it is always extensive it is also extremly small and the leading extensive part decays very fast to zero as temperature is increased. 
Xiao-Hong Wang, Shlomo Havlin, Moshe Schwartz