The Wigner semicircle distribution takes a form that goes to zero beyond a certain R. I don't see why it can actually hit zero however. According to Wikipedia's article on the Wigner semicircle distribution,
This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity.
This seems to be saying that it is impossible to construct a random symmetric matrix that has arbitrary eigenvalues, since the probability of an eigenvalue taking a value outside the Wigner semicircle is zero. But that seems wrong, for example this symmetric matrix has the arbitrary eigenvalue $\lambda$:
$$\begin{pmatrix} \lambda & 0 \\ 0 & 0 \end{pmatrix}$$
It could be that a random symmetric matrix has a very low probability of taking this form, but a low probability still isn't zero.
What am I missing?