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An induced real quaternion spherical ensemble of random matrices

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a $2\times 2$ complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to $$\frac{\det (\mathbf{G} \mathbf{G}^{\dagger})^{2 L}} {\det(1_N+ \mathbf{G} \mathbf{G}^{\dagger})^{2(n+N+L)}}.$$ These matrices can also be constructed via a procedure called `inducing', using a product of a Wishart matrix (with parameters $n,N$) and a rectangular Ginibre matrix of size $(N+L)\times N$. The inducing procedure imposes a repulsion of eigenvalues from $0$ and $\infty$ in the complex plane, with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of $L$, $n$ and $N$. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue $m$-point correlation functions, and in particular the eigenvalue density (given by $m=1$). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of $n$ and $L$. After a stereographic projection the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behaviour of the density near the real line based on analogous results in the $β=1$ and $β=2$ ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the $β=4$ induced spherical ensemble.