What is known about the density of states of the Anderson model?

This question was posted a week ago on MathOverflow without an answer:
https://mathoverflow.net/questions/369156/what-is-known-about-the-density-of-states-for-the-anderson-model

The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(mathbb{Z}^d)$)

$$
H_omega = – Delta + V(omega)
$$
where $V(omega) mid x rangle = omega(x) mid x rangle$ with ${ omega(x) }_{x in mathbb{Z}^d}$ independent and uniformly distributed in $[-L, L]$.

It is known for example by Kunz and Soulliard that the spectrum $ sigma(H_omega) = [-L, 4d +L ]$. This is also easily verified by simulating finite dimensional approximations. By simulation it is also easy to look at the density of states. Below I have plotted the density of states for $L=0,1,5$ for a $500 times 500$ approximation. One can see that the high probabilities towards the end of the spectrum wash out. So far this is all numerics, but what is known analytically about this particular density of states?

<img src="https://i.stack.imgur.com/tGLos.png" alt="The density of states plotted for $L=0,1,5$ for 500×500 approximatio” />