Concentration of measure
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
A sphere that is nearly all equator
On an ordinary globe, the band within a tenth of the radius of the equator holds a tenth of the surface. On a sphere in a thousand dimensions the same band holds 99.85% of it, and the band of a fifth holds all but about two parts in ten billion. Almost every point of a high-dimensional sphere is near every equator at once — and so any function that cannot change quickly is, over almost all of the sphere, almost constant.
Edges that crowd each other out
Pick a spanning tree of a graph uniformly at random, and the presence of one edge makes every other edge less likely — a theorem of Kirchhoff's electricity. Pick a forest instead, or a connected subgraph, and the same is believed and unproved. Checking every pair of edges in every labelled graph up to six vertices, 871,926 pairs for forests alone, finds no exception, and comes as close as 0.9994 to one.
The longest thread between two random strings
Write down two random strings of a thousand bits each and find the longest sequence that can be read in both, skipping freely. It is about 81% of each, and the exact share — the Chvátal–Sankoff constant — has resisted computation for fifty years. Its fluctuations are a smaller mystery with a sharper edge: the variance is known to be at most n/2, and over every length that can be computed it grows like n to the power 0.7, with nobody able to say what it does after that.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchVarianceConvergence rateCorrelationCurse of dimensionalityDimensionExpectationIsoperimetric inequalityLipschitz functionNegative correlationNormal distributionOpen problem