The directed polymer in a random environment

A continuous space/time approximation of the well known ‘directed polymer’ problem is considered. Connection between the ‘Helmholtz Free Energy’ and the ‘Two Walker problem’ is shown. Rigorous proof of the superdiffusive mean squared displacement exponent of 4/3 is given when there is one space dimension and one time dimension. Asymptotically diffusive behaviour of c(k)tis shown when there are one ‘time’ and two ‘space’ dimensions. For higher dimensions, the behaviour is diffusive and the mean squared displacement is asymptotically t d. These results hold for all temperature, because the phase transition in the discrete model is no longer present in the continuous model; the renormalization procedure has set the transition temperature to k _crit =0The joint distribution is also shown to be asymptotically sub-Gaussian for all dimensions and all temperatures (in the sense that the p ^thmoments as a function of pincrease more slowly than the moments of a Gaussian distribution). The ‘Helmholtz Free Energy’ is also calculated for this model and the quenched and annealed free energies are shown to be identical for all temperature