Concept

Random matrix

A matrix whose entries are drawn at random, studied through the statistics of its eigenvalues. Large ones obey universal laws, such as the semicircle and repulsion between neighbouring eigenvalues, that do not depend on the entries' distribution.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

400 random points and their longest rising chain, 34 long. 400 uniform random points in a unit square with the longest chain increasing in both coordinates marked: 34 points, against 2√n = 40.0.

The longest climb of a shuffle

Erdős and Szekeres guarantee that any n distinct numbers hold a climb or a fall of √n, and there are orders that allow nothing more. Shuffle a deck at random instead and the longest climb is almost exactly 2√n — with fluctuations of size n to the power one-sixth, distributed exactly as the largest eigenvalue of a large random matrix.

discrete · Ramsey theory

Eigenvalues that will not crowd together. Unfolded spacings, eigenvalues: 0.36 1.52 1.15 0.88 1.23 1.13 0.34 0.75 2.05 0.65 1.66 0.64 1.07 1.01 0.86 1.23 0.69 1.30 0.77 0.94 0.99 0.71 0.92 2.49 0.32 1.07 0.62 1.34 0.07 0.51 0.98 2.25 0.64 0.78 0.35 0.95 1.36 1.89 0.21 0.79 0.46 0.84 1.12 1.52 0.96 0.65 1.09 0.29 1.30 1.06 1.83 1.50 0.28 0.43 1.19 1.01 0.52 1.54 1.51 1.56. Independent points: 0.34 0.01 0.31 3.72 0.25 2.29 0.98 2.14 1.99 0.54 0.15 0.58 0.96 0.22 0.26 1.11 0.04 0.38 0.55 0.74 0.89 0.11 1.47 0.52 0.06 2.11 1.29 3.04 0.44 0.28 2.49 3.97 0.87 1.04 0.83 0.77 0.79 0.69 1.76 1.82 0.14 0.07 1.03 0.88 1.05 0.37 0.06 0.82 0.70 1.72 0.90 1.24 0.32 0.25 1.07 0.55 1.59 3.23 0.36 0.40. Gaps under a quarter of the mean: 2 against 9.

Eigenvalues that keep their distance

Fill a large symmetric matrix with independent random numbers and its eigenvalues do something no list of independent numbers does: they refuse to crowd together. The reason is a count of conditions in a two-by-two matrix, and its consequences — a semicircle, gaps that vanish at nought, a spectrum almost as rigid as a lattice — reach from heavy nuclei to the zeros of the zeta function.

algebra · Eigenvectors

Named alongside it

The objects these essays reach for when they reach for this one.

Avoided crossing Eigenvalue Fluctuations Independence Interlacing Limit shape Longest increasing subsequence Monte Carlo Random permutation Robinson schensted Semicircle law Symmetric matrix

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