Random Matrix Theory & the Semicircle Law
Eigenvalue distributions of large random matrices — and why they're universal.
Take an $N \times N$ symmetric matrix with i.i.d. entries $X_{ij}$ above the diagonal (zero mean, variance $1$) and $X_{ii}$ (zero mean, variance $2$). As $N \to \infty$, the empirical eigenvalue distribution of $X/\sqrt N$ converges to Wigner's semicircle:
$$\rho(\lambda) = \frac{1}{2\pi} \sqrt{4 - \lambda^2}, \quad |\lambda| \leq 2.$$Three canonical ensembles distinguished by symmetry of entries:
- GOE (Gaussian orthogonal): real symmetric, time-reversal symmetric Hamiltonians.
- GUE (Gaussian unitary): Hermitian, no time-reversal.
- GSE (Gaussian symplectic): self-dual quaternionic, half-integer spin + TR.
Wigner surmise: nearest-neighbor eigenvalue spacing distribution depends on the symmetry class — universal level repulsion at small $s$. RMT explains spectra of complex nuclei, chaotic billiards (Bohigas–Giannoni–Schmit conjecture), and zeros of the Riemann zeta function (Montgomery–Dyson).
For non-Hermitian matrices: the circular law says eigenvalues fill the unit disk uniformly. For products of random matrices: Lyapunov exponents; for sample covariance: Marchenko–Pastur. RMT is a rich modern subject with applications from QCD to wireless communications and machine learning.