The random model:

On , consider the random operator



where

 is compactly supported,
 are real, bounded, i.i.d. random variables.

Define the integrated density of states: 
let  be a large box




The limit exists almost surely under quite general assumptions.
Throughout the talk we assume it exists for the model we consider
It is non-decreasing. 
Under general assumptions (ergodicity), it is non-random.