free probability and freeness
Free probability is a non-commutative probability theory invented by Voiculescu, originally for operator algebras, that turned out to be the natural language for the asymptotic spectra of large random matrices. The core insight is a profound analogy: large random matrices behave, in their large-N limit, like 'free' non-commuting random variables, and freeness is to non-commutative variables what independence is to ordinary ones. Where classical probability has independence and the convolution of distributions, free probability has freeness and the free convolution — and it is freeness, not independence, that governs how the spectrum of A + B relates to the spectra of two large random matrices A and B.
The setting is a non-commutative probability space: an algebra of (non-commuting) elements with a linear 'expectation' functional tau, the trace, satisfying tau(1) = 1. For an N-by-N matrix, tau is the normalised trace (1/N) trace, and the 'moments' tau(a^k) of an element a are exactly the moments of its eigenvalue distribution. Two subalgebras (or two elements a and b) are called free if, whenever you take an alternating product of centred elements — p_1(a) q_1(b) p_2(a) q_2(b) ... where each factor has been adjusted to have tau equal to 0 — the trace of the whole product vanishes: tau(centred alternating product) = 0. This is a precise non-commutative rule that mimics, but is genuinely different from, the factorisation tau(ab) = tau(a) tau(b) you would get from independence. Crucially it accounts for non-commutativity: freeness determines all mixed moments tau(a^{k_1} b^{m_1} a^{k_2} ...) from the individual moments of a and b alone, just as independence does classically — but by a different, order-sensitive formula. The combinatorial heart of freeness is non-crossing partitions: where classical independence and cumulants live over all partitions, free cumulants live over non-crossing partitions only, the very same objects that counted the semicircle's moments.
Free probability matters because it gives an algebra for spectra. If A_N and B_N are large independent random matrices and at least one is unitarily invariant (so their eigenvectors are in 'generic position' relative to each other), then in the limit A_N and B_N become free, and the limiting eigenvalue distribution of A_N + B_N is the free additive convolution of their individual limits — computable, not by classical convolution, but by the R-transform. The free analogue of the Gaussian is the semicircle: a sum of many free copies, suitably normalised, converges to a semicircular element (the free central limit theorem). This is why the semicircle is to free probability what the Gaussian is to classical probability. A caveat: freeness is an asymptotic, large-N statement about the eigenvectors being in generic relative position; two fixed deterministic matrices are essentially never free, and if two random matrices share eigenvectors (e.g. they commute) freeness fails completely and you are back to classical convolution of their joint spectrum.
Take two large GUE matrices A_N and B_N, independent, each with a semicircle spectrum on [-2, 2]. Their eigenvectors are in generic relative position, so in the limit they are free. What is the spectrum of A_N + B_N? Not the classical convolution of two semicircles. By the free CLT a sum of free semicirculars is again semicircular: A_N + B_N has, in the limit, a semicircle spectrum on [-2 sqrt(2), 2 sqrt(2)] — variances add, exactly as classical variances add for sums of independent Gaussians, but the law stays semicircular, not Gaussian.
Freeness, not independence, governs the spectrum of a sum of large random matrices; the semicircle is its 'Gaussian'.
Freeness is an asymptotic statement requiring eigenvectors in generic relative position (one matrix unitarily invariant). Commuting or eigenvector-sharing matrices are NOT free; for them the spectrum of A + B is the classical convolution of their joint eigenvalues, not the free one.