Past exam of the random matrix theory course of the University of Oxford / 2025 / 1 / b / i / Solution

The spectral theorem for real symmetric matrices identifies the sum of the $k$th powers of the eigenvalues with the trace of the $k$th matrix power. Therefore the moment of the empirical spectral measure isFor odd $k$, every summand has zero expectation by part (a)(ii), so $\boxed{\mathbb{E}[m_k(n)]=0}$ for every $n$.

For the second moment, use $M_{ji}=M_{ij}$. There are $n$ diagonal entries with variance two and $n(n-1)$ ordered off-diagonal entries with variance one. Their means vanish, givingHence $\boxed{\underset{n\to \infty }{\lim }\mathbb{E}[m_2(n)]=1}$.