Theory of avoided criticality in quantum motion in a random potential in high dimensions

The density of states of a three-dimensional Dirac equation with a random potential as well as in other problems of quantum motion in a random potential placed in sufficiently high spatial dimensionality appears to be singular at a certain critical disorder strength. This was seen numerically in a variety of studies as well as supported by detailed renormalization group calculations. At the same time it was suggested by a number of arguments accompanied by detailed numerical simulations that this singularity is rounded off by the rare region fluctuations of random potential, and that tuning the disorder past its critical value is not a genuine phase transition but rather a crossover. Here we develop an analytic theory which explains how rare region effects indeed lead to rounding off of the singularity and to the crossover replacing the transition.