A real $2n\times 2n$ Hamiltonian matrix has the general form


$$H=\begin{pmatrix}
A & B \cr
C & -A^T
\end{pmatrix}
$$


where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are symmetric. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above condition is equivalent to the symmetry of $JH$ with


$$
J=\begin{pmatrix}
0 & 1 \cr
-1 & 0
\end{pmatrix},
$$


so it would be natural to consider the Gaussian orthogonal ensemble for $JH$. Generically, the eigenvalues come in quadruples: $\lambda, -\lambda, \bar\lambda, -\bar\lambda$



Eigenvalues of random Hamiltonian matrices
In the course of a [physics project](https://arxiv.org/abs/1305.2924) in my group, I have had an opportunity to learn more about the eigenvalue statistics of Hamiltonian matrices. (Our physics problem actually involved skew-Hamiltonian matrices, so I made a small detour, joined by Jonathan Edge & Jan Dahlhaus.)


The ensemble is the one you suggested: $2n\times 2n$ real matrices $H$ with Hamiltonian symmetry and normally distributed elements. It is convenient to rescale the eigenvalues $\varepsilon\_k$ of $H$ by a factor $\sqrt{2n}$, and separate the real and imaginary parts:


$(2n)^{-1/2}\varepsilon\_k=x\_k+iy\_k$.


The eigenvalue density in the complex plane $x+iy$ consists of three parts: a two-dimensional density $\rho\_{c}(x,y)$ of the complex eigenvalues, a one-dimensional density $\rho\_{r}(x)$ of the real eigenvalues and another one-dimensional density $\rho\_{i}(y)$ of the imaginary eigenvalues.


Based on numerical experiments, I can offer three conjectures:


1) For large $n$, the rescaled complex eigenvalues $x\_k+iy\_k$ uniformly cover a disc of unit radius, 


$\lim\_{n\rightarrow\infty}n^{-1}\rho\_{c}(x,y)=2/\pi$ for $x^2+y^2<1$.


2) For large $n$, the rescaled real eigenvalues $x\_k$ uniformly cover the interval $-1<x<1$, with density 


$\lim\_{n\rightarrow\infty}n^{-1/2}\rho\_{r}(x)=1/\sqrt{\pi}$. 


Therefore the expectation value of the number $n\_{r}$ of real eigenvalues satisfies $\lim\_{n\rightarrow\infty}n^{-1/2}E[n\_{r}]=2/\sqrt{\pi}$.


3) Also the rescaled imaginary eigenvalues $iy\_{k}$ have a uniform density in the large-$n$ limit, in the interval $-1<y<1$, but this density is less than the density of the real eigenvalues. The expectation value of the number $n\_{i}$ of imaginary eigenvalues satisfies $\lim\_{n\rightarrow\infty}n^{-1/2}E[n\_{i}]={\rm constant}\approx 0.72$.


Conjectures 1 and 2 were [proven](https://www-math.mit.edu/~edelman/homepage/papers/circular.pdf) by Edelman and collaborators in the absence of Hamiltonian symmetry, so when all $(2n)^{2}$ real matrix elements of $H$ are chosen from independent normal distributions. [This is known as the (real) Ginibre ensemble.] Our numerics suggests that, for large matrices, the Hamiltonian symmetry only affects the (rescaled) eigenvalue distribution within a distance of order $n^{-1/2}$ from the imaginary axis.


By way of illustration, I include a plot of the eigenvalues $\varepsilon$ of $200$ real matrices of size $100\times 100$ (so $n=50$), with normally distributed matrix elements, both with the Hamiltonian symmetry (left) and without (right). These eigenvalues are shown without rescaling, so they cover a disc of radius $\sqrt{2n}=10$.


![](https://www.lorentz.leidenuniv.nl/beenakker/MO/Hamiltonian_Ginibre.png)


