the circular law
The circular law is what happens to eigenvalues when you drop the symmetry. A Wigner matrix is forced to be Hermitian, so its eigenvalues are real and lie on a line (the semicircle). But the most naive random matrix of all — fill every entry independently, with no symmetry constraint whatsoever — has complex eigenvalues, and they do not fill a line: they fill a disc, uniformly. This is the circular law, the non-Hermitian counterpart of the semicircle, and proving it rigorously turned out to be dramatically harder than the Hermitian case.
Let A_N be an N-by-N matrix with iid entries of mean 0 and variance 1, with no symmetry imposed (so the entry above and below the diagonal are independent, not equal). Normalise to H_N = A_N / sqrt(N). The N eigenvalues of H_N are now complex numbers scattered in the plane. The circular law states that the empirical spectral distribution of these complex eigenvalues converges to the uniform probability measure on the unit disc {z in C : |z| <= 1} — constant density 1/pi inside the disc, zero outside. The reason this is hard, unlike the semicircle, is that for a non-normal matrix the eigenvalues are extraordinarily unstable under perturbation (the resolvent (H_N - z)^(-1) can be huge even far from any eigenvalue, the phenomenon of pseudospectrum). The successful proof, completed by Tao and Vu (building on Girko's strategy and Bai's work), goes through the Hermitization trick: study the eigenvalues of A_N via the singular values of the shifted matrix A_N - z I for each complex z, because log|det(A_N - z)| = sum of log of the singular values, and the logarithmic potential of the eigenvalue measure is exactly this average log-determinant. The hard analytic input is controlling the smallest singular value of A_N - z (the least-singular-value problem) so the logarithm is integrable.
The circular law matters as the universality statement for the most general iid matrix model, and it is the right null model whenever the relevant matrix is genuinely non-symmetric — the Jacobian of a large random dynamical system, the connectivity matrix of a neural network, an asymmetric interaction matrix in theoretical ecology (May's stability analysis: a large random ecosystem is stable only if its interaction strength stays below the radius of the eigenvalue disc). A caveat: the circular law describes the bulk of the complex spectrum and is universal in the entry distribution (finite variance suffices). But it is silent about the spectral radius — the single eigenvalue of largest modulus, which sits near the boundary |z| = 1 but with its own fluctuations — and it requires the entries to be genuinely independent with finite variance; correlations between symmetric pairs interpolate toward the elliptic law, and heavy tails break it.
A theoretical ecologist models S interacting species by a random S-by-S interaction matrix with iid entries of variance sigma^2, perturbing a stable diagonal. The eigenvalues fill a disc of radius sigma sqrt(S) (the circular law, rescaled). The ecosystem is stable only if this disc stays in the left half-plane, i.e. sigma sqrt(S) < 1: 'complexity' (large S or strong interactions sigma) destabilises. Robert May derived exactly this threshold in 1972 — the circular law is its rigorous backbone.
Drop symmetry and eigenvalues fill a disc, not a line; stability needs the disc inside the left half-plane.
The circular law is genuinely harder than the semicircle because non-Hermitian eigenvalues are unstable to perturbation (the pseudospectrum). The proof needs control of the smallest singular value of A_N - z, a deep input the Hermitian moment method never required.