Actually, just to get the semicircle is not hard. Take an $n$-by-$n$ Jacobi matrix whose on diagonal entries are $0$ and $i$th entry on the off diagonal is $\sqrt{i/n}$. The limit ESD will be the semi-circle.

The reason this work is that what you would have constructed is the mean part of the Dumitriu-Edelman Jacobi model for G$\beta$E, which will have the same limit for the empirical density of state. See https://arxiv.org/abs/math-ph/0206043 or the journal publication for details.

Spacing distributions are a different matter, but you did not ask about those....

Actually, just to get the semicircle is not hard. Take an $n$-by-$n$ Jacobi matrix whose on diagonal entries are $0$ and $i$th entry on the off diagonal is $\sqrt{i/n}$. The limit ESD will be the semi-circle.

The reason this works is that what you would have constructed is the mean part of the Dumitriu-Edelman Jacobi model for G$\beta$E, which will have the same limit for the empirical density of state. See https://arxiv.org/abs/math-ph/0206043 or the journal publication for details.

Spacing distributions are a different matter, but you did not ask about those....