Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on Z(+)(d) where to each edge e is associated a random variable upsilon(e). A polymer configuration is represented by a directed path r and has a weight exp[-beta Sigma(c epsilon r) upsilon(e)], with beta=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations.