the Wigner semicircle law
/ Wigner = VIG-ner /
The semicircle law is the central limit theorem of random matrix theory — the universal limiting shape of the bulk eigenvalue density of a Wigner matrix. Just as the sum of many independent scalars, suitably normalised, converges to a Gaussian regardless of the summands' distribution, the eigenvalue histogram of a large symmetric matrix with independent entries converges, regardless of the entry distribution, to a single fixed curve: a semicircle. It answers the founding question of the field, posed by Wigner for nuclear physics: what is the typical density of energy levels of a complicated quantum system?
Let H_N be a Wigner matrix normalised so the off-diagonal entries have variance 1/N. Then as N tends to infinity the empirical spectral distribution of H_N converges weakly, almost surely, to the semicircle distribution with density rho(x) = (1/(2 pi)) sqrt(4 - x^2) for x in [-2, 2] and zero outside. The graph of rho is the upper half of a circle (well, an ellipse scaled to a semicircle) of radius 2 — hence the name. Two of its k-th moments are illuminating: the integral of x^2 rho(x) dx is 1 (so the average of lambda_i^2 matches the entry variance, as the trace identity demands), and the even moments are the Catalan numbers C_k = (1/(k+1)) binomial(2k, k), which count non-crossing pairings — the combinatorial signature that makes the moment-method proof work. Equivalently, its Stieltjes transform m(z) solves the quadratic m(z) = 1/(-z - m(z)), giving m(z) = (-z + sqrt(z^2 - 4))/2.
The semicircle law is universal: it depends only on the entry variance, not on the shape of the entry distribution (provided the variance is finite and a mild moment condition holds). This is why it is the random-matrix analogue of the CLT. Its role across science is enormous — it governs the eigenvalue density of large adjacency and Hamiltonian matrices, underlies free probability (a semicircular element is the free analogue of a Gaussian), and provides the null spectrum against which real data spectra are compared. An important caveat: the semicircle describes only the bulk. The largest eigenvalue does converge to the edge value 2, but its fluctuations are NOT semicircular and not Gaussian — they are Tracy-Widom, on the finer scale N^(-2/3). The semicircle is silent about the edge.
Compute the fourth moment of a normalised Wigner matrix: E[(1/N) trace(H_N^4)] -> the integral of x^4 against the semicircle = 2, the second Catalan number. By the moment method this 2 counts exactly the two non-crossing ways to pair up four steps of a closed walk; the one crossing pairing is suppressed by a factor 1/N and dies in the limit. Combinatorics and the curve match perfectly.
Even moments of the semicircle are Catalan numbers — the count of non-crossing pairings.
The semicircle governs the bulk only and is universal in the entry distribution. It is NOT Gaussian-shaped (it has hard edges at +/-2 and zero density beyond), and it says nothing about edge fluctuations, which are Tracy-Widom on scale N^(-2/3).