I am now reading a paper by Sommers, H. J., et al. *"Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898.*, it claims a mathematical statement (equation (2) in the paper) as following:

Given an ensemble of $N\times N$ real asymmetric random matrix $J_{ij}$ defined by a Gaussian distribution with zero mean and correlations

$N[J_{ij}^{2}]_J = 1$ and $N[J_{ij}J_{ji}]_J = \tau$ for $i \neq j$ and $-1\leq \tau \leq 1$

these correlations can be derived from a Gaussian measure

$P(J) \propto \exp\big[ -\frac{N}{2(1 - \tau^2)}\mathrm{Tr}(JJ^T - \tau JJ) \big]$, where $J_{ij}^{T} = J_{ji}$.

But I am not sure how this derivation to be calculated, and in addition it says this measure implies for the diagonal elements $N[J_{ii}^2]_J = 1 + \tau$. Could someone help me to complete the detailed derivation ?

In short for questions:

(1) Why Gaussian measure $P(J)$ has this specific formula shown above (e.g. trace etc.)? How it can be related or even calculated from the definition of multivariate normal distribution or matrix normal distribution $$p(\mathbf{X}\mid\mathbf{M}, \mathbf{U}, \mathbf{V}) = \frac{\exp\left( -\frac{1}{2} \, \mathrm{tr}\left[ \mathbf{V}^{-1} (\mathbf{X} - \mathbf{M})^{T} \mathbf{U}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |\mathbf{V}|^{n/2} |\mathbf{U}|^{p/2}}$$ as it is shown in the Wikipedia page.

(2) How to derive the correlations value (i.e. 1 and $\tau$) from the defined Gaussian measure ?

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