Matrix model for "$\beta$-Ginibre" ensembles

Question

A very well known result in random matrix theory is that there exists "nice" (i.e., with independent entries) tridiagonal matrix for the $\beta$-ensembles of random matrix theory $$\propto\prod_{i<j}|\lambda_j-\lambda_i|^\beta\exp\left(-\frac{\beta}{4}\sum_{i}\lambda_i^2\right),\qquad\lambda_i\in\mathbb R\tag{1}$$ extrapolating between the classical values of $\beta=1,2,4$ given by the GO/U/SE (see this paper by Edelman and Dumitriu). This discovery subsequently paved the way for many interesting results regarding the behaviour of particle systems described by $(1)$, such as those related to the stochastic Airy operator.

It is often said that, from the point of view of statistical physics, the interaction term $|\lambda_i-\lambda_j|$ in the joint density $(1)$ is not natural in a one-dimensional setting, and that a more "appropriate" model for this type of repulsion would be in two dimensions: the "$\beta$-Ginibre" ensemble, given by $$\propto\prod_{i<j}|\lambda_j-\lambda_i|^\beta\exp\left(-c(\beta)\sum_{i}|\lambda_i|^2\right),\qquad\lambda_i\in\mathbb C.\tag{2}$$

After searching in the literature, I cannot find "nice" matrix models for $(2)$ that would extend the case $\beta=2$ for i.i.d. complex Gaussian models (though, as noted by Carlo Beenakker, the density $(2)$ for $\beta=1,4$ is not given by the real or quaternion i.i.d. Gaussian model).

Is it known that such models exist, but are somehow not very interesting in terms of applications, or are there good reasons to believe that such a model does not exist? Of course we cannot expect a self-adjoint tridiagonal model, but it seems to me a natural question wether or not this can be done and if there might be interesting applications.

Answer 1

this is not an answer to the question "what matrix model produces the eigenvalue distribution (2)", but it does explain why (2) cannot be an extrapolation between the $\beta$-Ginibre ensembles.

The eigenvalue distribution in the Ginibre ensemble is remarkably complicated, it only has the simple form (2) for $\beta=2$, see equations 1,2,3 of Eigenvalue statistics of the real Ginibre ensemble. If $\beta=4$, instead of a factor $|\lambda_i-\lambda_j|^4$ the repulsion is of the form $|\lambda_i-\lambda_j|^2 |\lambda_i-\bar{\lambda}_j|^2$, and for $\beta=1$ one has three repulsion factors: one factor $|\lambda_i-\lambda_j|^2 |\lambda_i-\bar{\lambda}_j|^2$ between complex eigenvalues $\lambda_i$, one factor $(\mu_i-\mu_j)$ between real eigenvalues $\mu_i$, and one factor $|\lambda_i-\mu_j|^2$ between a real and a complex eigenvalue.

So you see, the "interpolating distribution" (2) is not a natural object for eigenvalue statistics in the complex plane.

Answer 2

To the best of my knowledge, no general random matrix construction exists for two-dimensional log-gases, i.e. nothing like Dumitriu-Edelman's result in one-dimension. However, it is possible to find models for specific $\beta\neq2$, e.g. Hastings [arXiv:cond-mat/9909234] claims to have a model for $\beta=4$ (as Carlo points out, this is different from the quaternionic Ginibre ensemble). There's a recent summary by Peter Forrester [arXiv:1511.02946] on "Analogies between random matrix ensembles and the one-component plasma in two-dimensions"; several relevant reference may be found within.