Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:

$\begin{align} &\left<A_{ij}\right>=p &\text{and } & \text{var}(A_{ij})=\frac{1}{N}p(1-p) &\text{ if } \, i=j \\ &\left<A_{ij}\right>=p^2 &\text{and } &\text{var}(A_{ij})=\frac{1}{N} p^2(1-p^2) &\text{ if } \, i\neq j \end{align}$

where $p \in (0,1)$ is some fixed paremeter. The matrix is symmetric $A_{ij}=A_{ji}$.

Now I would like to know

$\begin{align} \mathbb{E}[\lim_{N\rightarrow \infty} |\det A|], \end{align}$

i.e. the asymptotic behavior as $N$ becomes large. Since the variances are approaching zero for $N \rightarrow \infty$ I assume that the expectation of the absolute value of the determinant also approaches zero.

Is there a method to determine this behavior or does someone know how to do it?

If I may ignore the fluctuations for large $N$, I can use that the average matrix $\bar{A}=E(A)$ has $N-1$ eigenvalues equal to $p(1-p)$ and one eigenvalue equal to $p(1-p)+Np^2$. The determinant then decays exponentially as ${\rm det}\,\bar{A}\approx Np^{N+2}(1-p)^N$ for $N\gg 1$.

To assess the effect of the fluctuations I might consider the matrix $\delta A=A-\bar{A}$, where the matrix elements have zero average. The eigenvalues $\lambda_n$ of $\delta A$ are distributed according to the Wigner semicircle in the interval $|\lambda|<2p(1-p)$. That determinant can be at most $2^{-N}$, which also decays exponentially.

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