Random matrix theory

The trace of a high power of H sums over closed walks on the index set; because entries are independent with mean 0, only walks that traverse each edge an even number of times survive expectation, and counting the dominant surviving walks reduces to a purely combinatorial problem whose answer is the Catalan numbers — exactly the moments of the semicircle distribution.

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.

A 2x2 matrix is the smallest system that has a gap between two eigenvalues at all, and it is small enough to compute the exact joint density of its entries by hand — yet it already contains the essential mechanism (the gap depends on an off-diagonal entry that must vanish for a degenerate eigenvalue, and vanishing is a single extra condition, which is what produces repulsion instead of clustering).

Step 1 (setup). Let $H = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ be a $2\times2$ GOE matrix with $a,c$ independent standard normal and $b$ independent normal with variance $\frac12$. Its characteristic polynomial gives eigenvalues $\lambda_{\pm} = \frac{a+c}{2} \pm \frac{s}{2}$ where the gap is $s = \sqrt{(a-c)^2 + 4b^2}$.

Step 2 (change of variables). Set $u = \frac{a-c}{\sqrt{2}}, \; v = b\sqrt{2}$; a direct check of variances shows $u$ and $v$ are independent standard normal variables, and $s = \sqrt{u^2+v^2}$ exactly, since $(a-c)^2+4b^2 = 2u^2+2v^2$ simplifies to $2(u^2+v^2)$ after the substitution — so $s = \sqrt{2}\cdot\sqrt{(u^2+v^2)/2}$, i.e. $s$ is $\sqrt{2}$ times the radius of a standard 2D Gaussian vector.

Step 3 (polar coordinates). The radius $r=\sqrt{u^2+v^2}$ of a standard 2D Gaussian vector is Rayleigh distributed with density $r\,e^{-r^2/2}$ (from integrating the joint Gaussian density $\frac{1}{2\pi}e^{-(u^2+v^2)/2}$ over the angle, which contributes a factor $2\pi$, and the Jacobian $r$ from $du\,dv = r\,dr\,d\theta$). Substituting $r = s/\sqrt2$ and using $dr = ds/\sqrt2$ turns this into a density $c\cdot s\, e^{-s^2/4}$ in $s$ for some constant $c$.

Step 4 (normalize). Fixing $c$ so that the mean spacing $\int_0^\infty s\cdot p(s)\,ds = 1$ (the convention used throughout the theory) pins down $c = \pi/2$, giving exactly $p(s) = \frac{\pi s}{2} e^{-\pi s^2/4}$ — matching the claimed formula.