The eigenvalue/singular values of (large) square random matrices $M$ is an iid random matrix with $M_{ij} \sim \mathcal{N}(0,\frac{g^2}N)$ except that the diagonal entries are $-1$.
I am to compute, in the limit $N\to\infty$,

*

*the eigenvalue/singular value spectrum/distribution. I.e. we order the eigenvalues according to their relevant values such that $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_N$, and then we hope to know the mean of each $\lambda_i$) of a random matrix $M$:
$$\bar \lambda_1, \bar\lambda_2, \dots, \bar \lambda_N$$ 

*the mean ($\ln$) thereof:
$$\langle \ln \lambda \rangle = \frac{\ln\lambda_1 + \ln\lambda_2 + \dots + \ln\lambda_N}N.$$
It seems we can calculate $\langle \ln \lambda \rangle$ using $\det(M)$. Then how can we compute the determinant (also a statistics) of such a random matrix?
We possibly need to use the Wigner semicircular law. Possible methods in random matrices or more general probability include

reductions, the Fourier method, the moment method, the Lindeberg swapping trick, individual  swapping, Stein's method, Predecessor comparison.

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I hope to know any books or papers about solving the above (or other) statistics of this (or simpler similar) random matrix, or of other 'symmetric' (w.r.t. probability distribution of each entry) matrices, or of other kinds of (large) square random matrices. The literature of random matrices is vast so I hope to narrow down the scope of my research. I can read them to find clues. Exact solutions are welcome but not necessarily expected right now.

(This section is unnecessary for answering the question:) An example of calculation is given as (note that $\mu_i$ below is actually eigenvalues denoted as $\lambda_i$ above. I hope to know what is more general than the example):




This seems to be a seminal paper in the topic:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.60.1895
A closely related post: Spectrum of large random asymmetric matrices with correlation
 A: If I slightly modify your prescription, taking i.i.d. Gaussians (mean zero, variance $g^2/N$) for all matrix elements, including the diagonal elements, then your matrix is a member of the (real) Ginibre ensemble; for large $N$ this modification only subtracts 1 from each eigenvalue.
The density of the eigenvalues in the Ginibre ensemble is given by Girko's circular law, and the density of the singular values is given by the Marchenko-Pastur quarter circular law, see for example Around the circular law. These densities refer to the bulk of the spectrum, in addition there are of order $\sqrt N$ real eigenvalues, distributed uniformly in the interval $(-1,1)$. This line of real eigenvalues is clearly visible in the numerical data shown below (for $N=100$); notice that the line is surrounded by a narrow depletion region, since the complex eigenvalues are repelled from the real axis. See the plot below.

Eigenvalues in the complex plane of 200 real matrices of size 100 × 100 in the Ginibre ensemble, from https://arxiv.org/abs/1305.2924 
