# Probability of pr .: expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements

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= p & text {y} & text {var} (A_ {ij}) = frac {1} {N} p (1-p) & text {if} , i = j \ Y= p ^ 2 & text {y} & text {var} (A_ {ij}) = frac {1} {N} p ^ 2 (1-p ^ 2) & text {if} , i neq j 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?