Localization with off-diagonal disorder: A qualitative theory

I present a theory of localization for Hamiltonians with off-diagonal disorder in general dimension $d$. At $d=2\mathrm{}$ for weak disorder the $E=0\mathrm{}$ eigenstate decays as $\exp [-\gamma {\left(\ln N\right)}^{\frac{1}{2}}]$, with the coefficient $\gamma$ varying with disorder. For strong disorder in all dimensions there is weak localization at $E=0\mathrm{}$ and associated anomalies for $E\to 0$: the local density of states and rate of exponential decay are of the form found by Dyson in one dimension.