Spherical Matrix Ensembles

The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_{\beta }(N,r)$ consist of $N\times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N\to \mathrm{\infty }$. In the unitary case ($\beta =2$), we also find an explicit formula for the empirical spectral density for each $N$.