Wigner's semicircle distribution is:

$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$

Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows this distribution, from which we can infer both the number of rescaled eigenvalues in an interval $[a,b]$ and roughly where the $i$'th eigenvalue lies. Specifically, this concerns the statistics of the (unordered) vector $(\lambda_1(M),\lambda_2(M),\cdots,\lambda_n(M))$.

My question is, *where else* does the Wigner semicircle law arise? To push the discussion in a particular direction, I am primarily interested in other random processes $(X_1,\cdots,X_n)$ which asymptotically have semicircle statistics similar to the above. **Specifically, I am not interested in processes which can be somehow coupled to random matrices**. I would love to see some examples where the semicircle law occurs more or less universally, that is, for a wide, reasonable class of different distributions on $X_i$'s.

wouldbe the kind of thing the OP wants? $\endgroup$