Schr\"odinger operators with random $\delta$ magnetic fields

We shall consider the Schr"odinger operators on ${\mathbf{R}}^2$ with random $\delta$ magnetic fields. Under some mild conditions on the positions and the fluxes of the $\delta$-fields, we prove the spectrum coincides with $[0,\mathrm{\infty })$ and the integrated density of states (IDS) decays exponentially at the bottom of the spectrum (Lifshitz tail), by using the Hardy type inequality by Laptev-Weidl. We also give a lower bound for IDS at the bottom of the spectrum.