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

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 $\boxed{\underset{n\to \infty }{\lim }\mathbb{E}[m_2(n)]=1}$.

Write the four cyclic factors as $A=M_{ij}$, $B=M_{jk}$, $C=M_{kl}$ and $D=M_{li}$. The Gaussian moment pairing theorem gives three terms, corresponding to $(AB)(CD)$, $(AC)(BD)$ and $(AD)(BC)$. We sum each term over $i,j,k,l$.

For $(AB)(CD)$, the entry covariance givesIndeed the four products leave respectively three, two, two and one free indices. A cyclic relabelling gives exactly the same sum for $(AD)(BC)$.

For the crossing pairing $(AC)(BD)$,The first product allows $i=k$ and $j=l$ freely, whereas each other product forces all four indices to coincide. Combining the three pairings and the trace normalization,