  The nonlinear Hall effect has recently attracted significant interest due to
its potential as a promising spectral tool and device applications. A theory of
the nonlinear Hall effect on a disordered lattice is a crucial step towards
explorations in realistic devices, but has not been addressed. We study the
nonlinear Hall response on a lattice, which allows us to introduce strong
disorder numerically. We reveal a disorder-induced fluctuation of the Berry
curvature that was not discovered in the previous perturbation theories. The
fluctuating Berry curvature induces a fluctuation of the nonlinear Hall
conductivity, which anomalously increases as the Fermi energy moves from the
band edges to higher energies. More importantly, the fluctuation may explain
those observations in the recent experiments. We also discover an "Anderson
localization" of the nonlinear Hall effect. This work shows a territory of the
nonlinear Hall effect yet to be explored.
