I am trying to determine the determinant of the following uniformly distributed random symmetric matrix $A$ with zero mean.
\begin{equation} A= \begin{pmatrix} 1 & \cos \alpha_{12} & \cos \alpha_{13} & \dots &\cos \alpha_{1N} \\ \cos \alpha_{12}& 1 & \cos \alpha_{23} & \dots &\cos \alpha_{2N} \\ \cos \alpha_{13} & \cos \alpha_{23} & 1 & \dots &\cos \alpha_{3N} \\ \vdots & \vdots & \vdots & \quad & \vdots \\ \cos \alpha_{1N} & \cos \alpha_{2N} & \cos \alpha_{3N} & \dots & 1\\ \end{pmatrix} \end{equation} Where each vectors are linearly independent and $\alpha_{ij} \in [-0.5,0.5]$, for all $i,j=1,2,3,\dots,N$.
Any valuable resource or help is appreciated.
Numerical Results: So far I have the following results:
As $N$ goes to infinity, the expected value of the determinant approaches zero. For instance, computing $k\times k$ determinants, for $2<=k<=N(=30)$ i.e. for $k=2$,
\begin{equation} A= \begin{pmatrix} 1 & \cos \alpha_{12} & \cos \alpha_{13} & \dots &\cos \alpha_{1N} \\ \cos \alpha_{12}& 1 & \cos \alpha_{23} & \dots &\cos \alpha_{2N} \\ \end{pmatrix} \end{equation} then $AA^T$ will be a $2\times 2$ matrix. This gives the following result. Expected value of the determinant for $2<=k<=30$The value of the determinant for a particular value $k$ increases as N goes to infinity. For example when $K=2$, the determinant of the $2\times 2$ matrix for N=10,20 and 30 respectively is $0.05210<0.05301<0.05329$.
Analytical Explanation I am looking for an analytical explanation to support the above numerical result.