Suppose that $\alpha_{1},\dots,\alpha_{r}$ are non-zero complex numbers. Let $U_{1},\dots,U_{r}$ be random $n\times n$-unitary matrices. Let $A=\alpha_{1}U_{1}+\dots+\alpha_{r}U_{r}$.
I have observed that if $r$ is much smaller than $n$, then the spectral radius of $A$ tends to be around $\sqrt{|\alpha_{1}|^{2}+\dots+|\alpha_{r}|^{2}}$. This is not too surprising to me since if $\mathbf{x}$ is randomly selected such that $\|\mathbf{x}\|=1$, then we usually have $$\|A\mathbf{x}\|\approx\sqrt{|\alpha_{1}|^{2}+\dots+|\alpha_{r}|^{2}}.$$ However, I noticed that when $r$ is much smaller than $n$, the eigenvalues of $A$ tend to be uniformly distributed on the disk $$\{z:|z|^{2}\leq|\alpha_{1}|^{2}+\dots+|\alpha_{r}|^{2}\}.$$
What is the explanation for this?
I will be interested in explanations of this phenomenon that generalize to the case where $U_{1}=\phi(g_{1}),\dots,U_{r}=\phi(g_{r})$ for randomly selected $g_{1},\dots,g_{r}\in G$ where $G$ is some finite or compact group and $\phi$ is some linear representation of $G$.
This question is motivated by block ciphers in cryptography. I want to contrast the behavior of the eigenvalues of $U_{1}+\dots+U_{r}$ where $U_{1},\dots,U_{r}$ are random permutation matrices with the eigenvalues of the eigenvalues of $V_{1}+\dots+V_{r}$ where $V_{1},\dots,V_{r}$ are the round permutations for some block cipher where the round key size is smaller than the block size.
