eigenvalue spacing and the sine kernel

If the semicircle law is the global picture of where eigenvalues are, eigenvalue spacing statistics are the local picture of how they are arranged relative to each other — the microscopic structure on the scale of a single eigenvalue gap. This is the level of detail that physicists actually cared about: Wigner's original motivation was the spacing between adjacent nuclear energy levels, and the spectacular empirical finding was that these spacings are NOT independent and exponentially distributed (as a Poisson process of independent points would give). They repel one another. The mathematical object that captures this repulsion in the limit is the sine kernel.

Zoom in on the eigenvalues near a fixed point E in the bulk of the spectrum and rescale so the mean spacing becomes 1 (multiply gaps by the local density times N). In this microscopic limit the eigenvalues form a determinantal point process whose correlation kernel is the sine kernel: K(x, y) = sin(pi (x - y)) / (pi (x - y)). 'Determinantal' means the probability of finding eigenvalues at positions x_1, ..., x_k is the determinant of the matrix [K(x_i, x_j)], and all the local statistics — the gap distribution, the variance of eigenvalue counts in an interval — follow from this single kernel. The defining consequence is level repulsion: the probability density of finding two eigenvalues at small separation s vanishes as s -> 0, for the GUE like s^2 (and like s^beta in general, beta = 1, 2, 4). Contrast this sharply with a Poisson process, where points are placed independently and the nearest-neighbour spacing is exponentially distributed with positive density at s = 0 — Poisson points cluster, eigenvalues avoid each other. A famous closed-form approximation, the Wigner surmise, gives the GUE gap density as approximately (32/pi^2) s^2 exp(-(4/pi) s^2), capturing the s^2 repulsion and the Gaussian-tailed decay.

The sine kernel is the universal local law of the bulk: bulk universality is the theorem that a generic Wigner matrix (any entry distribution with the right moments) exhibits sine-kernel statistics in the bulk, identical to the Gaussian ensemble of the same symmetry class. This universality is why random matrix spacing distributions are a genuine signature found across nature — in the resonance levels of heavy nuclei, the zeros of the Riemann zeta function (Montgomery-Odlyzko), the energy levels of chaotic quantum billiards, and even the spacing of parked cars and bus arrival times. The sine kernel is the fingerprint of correlated, repelling spectra. A caveat: level repulsion is a property of the local statistics, not of the global density; and the sine kernel is the BULK law. At the edge of the spectrum the correct kernel is the Airy kernel (whose largest-particle distribution is Tracy-Widom), and at a hard edge it is the Bessel kernel. One must say where in the spectrum one is looking.