Wei-Min Wang Mean field bounds on Lyapunov exponents in $Z^d$ at the critical energy (124K, dvi file) ABSTRACT. We further develop the supersymmetric formalism initiated in [W1] (see also [SjW]). We obtain the optimal mean field bounds at the critical energy for Lyapunov exponents of random walks in random potentials in $Z^d$ at weak disorder. This extends some of the results in [W1].