Eigenvalue distribution of real anti-symmetric Gaussian random matrices

Question

Let $A_{ij} = - A_{ji}$ be a $n \times n$ matrix with real entries distributed according to a Gaussian distribution with zero mean and standard deviation $\sigma$. What is the eigenvalue distribution of such matrices? I am mostly interested in the case where $n\rightarrow\infty$. If the matrices were symmetric, rather than anti-symmetric, then the answer to this question in the $n \rightarrow \infty$ limit would be the Wigner semi-circle law.

Answer 1

I can't find a reference but based on a few simulations the imaginary part of the eigenvalues (the real part is of-course zero) seem to follow the semi-circle law as well if the entries are picked independently from the standard normal distrbution. Here is a quick python script:

%matplotlib inline   # For use in a Jupyter notebook
import numpy as np 
import matplotlib.pyplot as plt

N=100
mu = 0
sigma = 1

matlist = list()
eval_list = list()

for i in range(1,1000):
    mat = sigma*np.random.randn(N,N) + mu
    matlist.append((mat - mat.T)/2)

for matrix in matlist:
    eigs, vecs = np.linalg.eig(matrix)
    eval_list.extend(map(lambda k: np.imag(k), eigs))

plt.figure()
plt.hist(eval_list)
plt.show()

Answer 2

For such random matrices $A$, we have that $iA$ is a complex Wigner matrix (i.e. hermitian, zero-mean, independent entries). So the spectrum of $iA$, which is $i$ times the spectrum of $A$, follows the semi-circle law