the Tracy-Widom distribution
/ Tracy-Widom = TRAY-see WID-um /
The Tracy-Widom distribution describes the fluctuations of the largest eigenvalue of a random matrix — the edge of the spectrum. The semicircle and Marchenko-Pastur laws tell you the largest eigenvalue converges to the upper edge of the bulk (2 for the Wigner case, (1 + sqrt(y))^2 for sample covariance), but they say nothing about how it fluctuates around that edge. Tracy-Widom is the answer, and it has turned out to be one of the most surprising universal laws in all of mathematics, appearing far beyond random matrices — in the longest increasing subsequence of a random permutation, in growing interfaces (KPZ universality), in queueing systems, and in directed polymers.
For an N-by-N GUE matrix with eigenvalues lambda_1 <= ... <= lambda_N, the largest eigenvalue lambda_N is approximately 2 for large N, but its precise fluctuations satisfy a limit theorem: N^(2/3) (lambda_N - 2) converges in distribution to the Tracy-Widom distribution. Two features deserve attention. First the scaling exponent: the edge fluctuations are of order N^(-2/3), much smaller than the bulk spacing N^(-1) is large — wait, more precisely, eigenvalues near the edge are spaced N^(-2/3) apart (sparser than the bulk N^(-1)), and the largest fluctuates on exactly that scale. Second the distribution itself: it is not Gaussian. It is given by the Tracy-Widom formula F_2(s) = exp(- integral from s to infinity of (x - s) q(x)^2 dx), where q is the Hastings-McLeod solution of the Painleve II equation q'' = s q + 2 q^3. The distribution is asymmetric, with a left tail decaying like exp(-|s|^3/12) (heavier, reflecting the elastic 'wall' of the other eigenvalues pushing in) and a right tail decaying like exp(-(4/3) s^(3/2)) (lighter). There are three versions — TW1, TW2, TW4 — for the three symmetry classes beta = 1, 2, 4.
Tracy-Widom matters because of its universality and its reach. Within random matrix theory, the edge fluctuations of a generic Wigner matrix (not just the exactly solvable Gaussian one) are Tracy-Widom — this is edge universality, a theorem requiring only matching of a couple of moments. Beyond random matrices, the same TW2 law governs the length of the longest increasing subsequence of a random permutation (Baik-Deift-Johansson), the height of a KPZ-class growing interface, and the largest particle in many determinantal point processes — making Tracy-Widom the 'central limit theorem' of a whole universality class of strongly correlated extremes. A crucial caveat: the N^(2/3) scaling and the TW law require the edge to be a 'soft edge' with a square-root vanishing density (the semicircle and MP both vanish like sqrt at their edges). A 'hard edge' (where the density blows up or is cut off, as at 0 for a square sample covariance matrix) has different, Bessel-kernel statistics, not Tracy-Widom.
Patience sorting a random shuffle of a deck. The length L_N of the longest increasing subsequence of a random permutation of {1, ..., N} satisfies (L_N - 2 sqrt(N)) / N^(1/6) converging to the Tracy-Widom TW2 distribution — the exact same law as the largest eigenvalue of a GUE matrix, with no matrix anywhere in sight. The appearance of the random-matrix edge law in a pure combinatorics problem (Baik-Deift-Johansson, 1999) was the discovery that opened up KPZ universality.
The same Tracy-Widom law governs the matrix edge and the longest increasing subsequence — a universality class.
Tracy-Widom is the soft-edge law, with the N^(2/3) scaling tied to a square-root-vanishing density. A hard edge (e.g. zero for a square Wishart matrix, where eigenvalues pile up) is governed by the Bessel kernel, not Tracy-Widom — match the edge type before invoking TW.