Bulk and edge universality

Fix an energy $E$ strictly inside the semicircle support. Since the empirical measure is normalized by dimension, an interval of length approximately $1/(N\rho_{\mathrm{sc}}(E))$ contains about one eigenvalue on average. Multiplying displacements by the reciprocal of this spacing unfolds the spectrum to unit intensity. For GUE the bulk kernel converges, after this rescaling, to the sine kernel. Its two-point factorial correlation at separation s is the displayed expression, giving repulsion near zero and a reference scale of order one.

The same formula is not the scalar correlation kernel for GOE. Real and complex symmetry have different local processes, although both share the semicircle global density. Nor does a limiting unit intensity mean every neighboring gap equals one. The point process has a distribution of gaps and correlated counts. A two-point correlation describes density of distinct pairs; interpreting it as a nearest-neighbor gap density would discard the condition that no intermediate eigenvalue lies between the two points.

One concrete non-Gaussian complex class takes independent upper entries $(a_{ij}+ib_{ij})/\sqrt N$, with all real variables a and b drawn independently from a fixed symmetric bounded continuous law of variance one half. Take independent real Gaussian diagonal entries of variance $1/N$, and fill the lower triangle by conjugation. The off-diagonal variance is then $1/N$ and the real and imaginary parts match. Bounded continuous laws supply subexponential tails and a strict fourth-moment gap; these are sufficient for the cited generalized-Wigner bulk theorem.

That theorem compares fixed-order correlation observables tested against continuous compactly supported functions, with a small energy-window average around an interior energy. First take matrix size to infinity at a fixed window, then shrink the window to zero. The resulting beta-two statistics agree with GUE and hence the sine process. This carefully stated version does not assert an unaveraged pointwise limit for every finite-moment model. Local laws, a short Gaussian regularization, and a comparison step are proof ingredients, each of which requires estimates beyond a global moment calculation.

Near the upper semicircle edge, the density vanishes like the square root of the distance to two. Integrating that density shows that a window containing order one eigenvalues has width of order $N^{-2/3}$, rather than the bulk order $N^{-1}$. Under standard normalized Gaussian ensembles, the largest eigenvalue centered at two and multiplied by $N^{2/3}$ converges to the Tracy–Widom law for the corresponding beta. This states a distributional limit, not a deterministic upper bound or the mean largest eigenvalue.

For a sufficient non-Gaussian real example, take independent centered bounded fixed-law upper off-diagonal variables of variance one before division by the square root of dimension, and independent centered Gaussian diagonal variables with fixed finite variance before that division. Lee and Yin’s edge theorem gives the beta-one Tracy–Widom limit. Bounded off-diagonal variables satisfy its required vanishing fourth-order tail condition. A model with occasional very large entries can violate that condition and change the extremes while retaining a global limiting law. Global convergence therefore cannot replace edge hypotheses.