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

$ begin {align}

Y

Y

end {align} $

where $ p in (0.1) $ It is a fixed paremeter. The matrix is symmetric. $ A_ {ij} = A_ {ji} $.

Now I would like to know

$ begin {align}

mathbb {E}[lim_{Nrightarrow infty} |det A|],

end {align} $

that is, the asymptotic behavior as $ N $ It becomes big. Since the variations approach zero for $ N rightarrow infty $ I suppose 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?