Fact 2: if the random variables admit a bounded density





Fact 3: the single-site perturbation is of rank one.



Plugging this back into the computation




This implies that the IDS is Lipshitz continuous.

Fact 2 : assumption on the model.

Fact 1 : if one moves the random parameters, the eigenvalues of the system move.

Fact 3 : local perturbation : it doesn't move too "many" eigenvalues at a time