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Concept

Percolation

A random model in which each edge or site of a grid is kept with some probability, and the question is whether the kept parts connect over long distances. Above a critical probability they do, and below it they do not.

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

The moment a giant piece appears. The largest component's share of 900 points plotted against the average degree, with the measured values as dots and the predicted curve behind them. The curve is flat at zero below an average degree of one and rises steeply above 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.

probability · Random graphs A ball of the square grid with random travel times. A ragged roughly round region of grid cells shaded in bands by the time a signal from the centre first reaches them.

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.

algebra · Cayley graph

Named alongside it

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

Random graphBranching processCayley graphComponentConvexityExpectationFirst-passage percolationFixed pointFluctuationsLimit shapePhase transitionSubadditivity

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