Given a $N \times M$, $N\ge M$ finite random matrix where the elements are drawn from a probability distribution with Kurtosis $\gamma$.

Is there anything that can be said about the singular values (or any other spectral property of either the matrix or its Gram matrix) as a function of $\gamma$? (if required, we can make simplified assumptions of zero-mean and other moments can be set arbitrarily).

Background: According to my numerical simulations, the conditioning of a random matrix seems to have a strong dependence on $\gamma$: The higher $\gamma$, the higher the condition number. Distributions with large kurtosis occur for example with polynomials of Gaussian distributions (as in my case).

I found A. Edelman et.al., Beyond Universality in Random Matrix Theory which initially looked promising. However, the results are for complex-valued matrices and furthermore, I cannot reproduce their results when I create distributions with high kurtosis (e.g. p=5;A = ((randn(n,n)).^p+1i*(randn(n,n)).^p).