Mean field upper and lower bounds on Lyapunov exponents

We extend the supersymmetric field theory formulation of random walks in random potentials developed in earlier papers. We obtain the mean field rate of decay for the squared Green's function at the critical energy. This requires studying supersymmetric field theories which are quartic in the Grassmann variables. We also obtain both the mean field upper and lower bounds at the critical energy for Lyapunov exponents for unbounded potentials. The upper bound part extends the corresponding result in an earlier work.