Wigner's semicircle law

Step 1 (target). The moments of the semicircle density $\rho(x) = \frac{1}{\pi}\sqrt{2-x^2}$ on $[-\sqrt{2}, \sqrt{2}]$ are the Catalan numbers: the $2k$-th moment equals $C_k = \frac{1}{k+1}\binom{2k}{k}$, and all odd moments vanish by symmetry. So it suffices to show the moments of the rescaled empirical eigenvalue distribution converge to these same numbers.

Step 2 (expand the trace). Write $\operatorname{Tr}(H^{2k}) = \sum_{i_1,\dots,i_{2k}} H_{i_1 i_2} H_{i_2 i_3} \cdots H_{i_{2k} i_1}$, a sum over closed walks of length $2k$ on $\{1,\dots,N\}$. Taking expectation and using independence of entries, $\mathbb{E}[\operatorname{Tr}(H^{2k})]$ splits into a sum, over ways of pairing up the $2k$ factors, of products of the second moments $\mathbb{E}[H_{ij}H_{kl}]$ of each pair (odd-order joint moments vanish for mean-zero, and unpaired factors vanish too since $\mathbb{E}[H_{ij}]=0$).

Step 3 (only non-crossing pairings survive at leading order). Each pairing corresponds to a way of identifying edges of the closed walk; a pairing contributes a factor of $N$ to a power determined by the number of distinct vertices visited. Counting shows a pairing contributes at order $N^{k+1}$ only when the identified edges form a non-crossing (planar) pairing of the $2k$ endpoints; crossing pairings contribute at strictly lower order in $N$ and vanish after dividing by $N^{k+1}$ to normalize.

Step 4 (count and conclude). The number of non-crossing pairings of $2k$ points on a circle is exactly the Catalan number $C_k = \frac{1}{k+1}\binom{2k}{k}$. Hence $\frac{1}{N}\mathbb{E}[\operatorname{Tr}((H/\sqrt{N})^{2k})] \to C_k$ as $N \to \infty$, matching the moments of $\rho(x) = \frac{1}{\pi}\sqrt{2-x^2}$ term by term; since the semicircle distribution is determined by its moments, the empirical spectral distribution converges to it.