TITLE: Semicircle law universality elsewhere
QUESTION [10 upvotes]: 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.

REPLY [4 votes]: Following up on the answers in the context of free probability theory one should note that there is actually a quite important and canonical operator on a Hilbert space having the semicircle as distribution. In the answers above there were already mentioned concrete operators arising from the free group representations, but there one also has to take the limit $d\to\infty$ to get the semicircle. I want to highlight a situation without taking a limit: Let $l$ be the one-sided shift (acting on a Hilbert space
with orthonormal basis $(e_i)_{i\geq 0}$ via $le_i=e_{i+1}$), then the selfadjoint operator $l+l^*$ has with respect to the "vacuum expectation" $\langle e_0, \cdot e_0\rangle$ the semicirlce law as distribution.
The one-sided shift is arguably one of the most important operators in single operator theory, but it seems nobody before Voiculescu was interested in looking at the distribution of its real part.