random-matrix universality
Universality is the deepest and most striking phenomenon in random matrix theory, and the reason the subject matters far beyond mathematics. The phenomenon is that the local statistics of eigenvalues — the spacing distribution in the bulk, the fluctuation of the largest eigenvalue at the edge — do not depend on the fine details of the matrix model. They depend only on the symmetry class (Hermitian, symmetric, or quaternionic, i.e. beta = 2, 1, 4) and not at all on the distribution of the individual entries. Gaussian entries, Bernoulli entries, any entries with the right first few moments: same local law. This is the matrix analogue, one level up, of the classical central limit theorem, where the limiting Gaussian forgets the distribution of the summands.
There are two faces, the global and the local, and universality refers chiefly to the local one. Global (or first-order) universality is the statement that the macroscopic eigenvalue density is the same across models — Wigner matrices give the semicircle, sample covariance matrices give Marchenko-Pastur, regardless of entry distribution; this is comparatively easy and follows from the moment or Stieltjes method using only the variance. Local universality is the deep statement: rescaled to the scale of individual eigenvalues, the correlation functions converge to the universal kernels of the matching Gaussian ensemble — the sine kernel in the bulk, the Airy kernel (Tracy-Widom) at the soft edge, the Bessel kernel at a hard edge. The modern proof strategy, developed by Erdos-Schlein-Yau and by Tao-Vu, has two ingredients: a local law controlling the resolvent down to the eigenvalue scale, and a comparison/interpolation argument. One route runs the Dyson Brownian motion (a stochastic flow of eigenvalues that relaxes any initial spectrum to the Gaussian local statistics in a short time), the other a four-moment theorem (the local statistics are unchanged if you match the first four moments of the entries to a Gaussian).
Universality is why random matrix theory is a predictive science rather than a study of one Gaussian model. Because the local laws are universal, they appear in systems that have nothing structurally to do with the Gaussian ensembles: the spacings of the nontrivial zeros of the Riemann zeta function follow the GUE sine kernel (Montgomery-Odlyzko), the energy levels of classically chaotic quantum systems follow GOE or GUE statistics (the Bohigas-Giannoni-Schmit conjecture), and the largest eigenvalue's Tracy-Widom law surfaces in growth models and combinatorics (KPZ). A caveat: universality is not unconditional. It requires the entries to be genuinely independent and to have enough finite moments (heavy-tailed entries with infinite variance break it and give a different, non-universal limit), and it requires the matrix to be in the right symmetry class with non-degenerate variance profile. Universality is a theorem with hypotheses, not a magic word — sparse or structured matrices (very sparse Erdos-Renyi graphs, band matrices below a threshold width) can leave the universality class entirely.
Montgomery computed the pair correlation of the nontrivial zeros of the Riemann zeta function (on the critical line, suitably unfolded) and found 1 - (sin(pi s)/(pi s))^2 — the exact pair correlation of the GUE sine kernel. Odlyzko then verified this numerically against billions of computed zeros to stunning precision. The zeta zeros, a deterministic arithmetic object, obey the same local repulsion law as the eigenvalues of a random Hermitian matrix. That coincidence, still unexplained at a deep level, is universality announcing itself in pure number theory.
The Riemann zeta zeros obey GUE sine-kernel statistics — universality reaching into number theory.
Universality holds only under genuine independence, enough finite moments, and the right symmetry class. Heavy tails (infinite variance), strong sparsity, or narrow band structure can push a matrix out of the universality class entirely — it is a theorem with hypotheses, not a slogan.