Consider $A$ to be a symmetric random matrix. When $i<j$, $A_{ij}=1+\sigma W_{ij}$ where $W_{ij}$ are standard gaussian variables, $A_{ij}=A_{ji}$ and $A_{ii}=\sum_jA_{ij}$. The $\sigma$ is a function on $n$.
My question is, what is the probability that the matrix $A$ is positive semi-definite?
Many thanks for any hint.