Percolation
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
The moment a giant appears
Raise the chance of an edge slowly and a random graph does nothing for a long time, then in a narrow window acquires a component holding a definite fraction of everything. The fraction is the root of an equation, and the equation says why the transition is where it is.
The shape a random ball grows into
Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.
The pieces no formula counts
A random vibration of a large membrane splits the surface into regions where it bulges up and regions where it dips down. Pieces minus holes has an exact average; the number of pieces has none. Nazarov and Sodin proved it is proportional to the area, Bogomolny and Schmit predicted the constant from percolation, 0.0624, and counting the pieces of 51 random waves gives 0.0599 — below the prediction by several standard errors. The sizes of the pieces follow percolation's law; their number does not.
Where random discs change sign
Scatter discs at random over the plane and their union breaks into pieces with holes in them. Pieces minus holes has an exact average, η e^(−η)(1 − η) per disc area, where η is how many discs cover a point on average, and it changes sign at exactly η = 1, with 63.2% of the plane covered. The discs first join up across the whole plane later, at η ≈ 1.128. The sign change looks like the moment the union connects and is not; and for squares or long rectangles turned at random it comes earlier still, at 4πA/L², which only a disc reaches one.
A board that cannot be drawn
Colour a rhombus of hexagons in two colours. One colour must join its two sides, and only one: in all 65,536 colourings of the four-by-four board, no exception. The proof is a walk along the colour boundary, the same walk that proves Sperner's lemma, and David Gale showed that the no-draw theorem of this board game is equivalent to Brouwer's fixed-point theorem. Nobody knows the winning strategy for a full-sized board, though a short argument proves one exists.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicFixed pointRandom graphThresholdBranching processBrouwerCayley graphComponentConnectednessConvexityCoverageCritical point