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Concept

Limit shape

The curve or region that a large random or combinatorial object settles onto once it is scaled down by its size. A ball in a lattice becomes a polygon, a random partition's diagram becomes a fixed curve, and the fluctuations about the limit are usually where the harder questions are.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Random partitions of 1,000, scaled, against their limit shape. The outlines of 4 uniformly random partitions of 1000, scaled by the square root of 1000, lying close to the curve e^(−cx) + e^(−cy) = 1 with c = π/√6.

The shape a random partition takes

There are about twenty-four thousand billion billion billion ways to write 1,000 as a sum of whole numbers. Pick one at random, draw its Ferrers diagram, shrink it by the square root of a thousand, and it is almost exactly the curve e^(−cx) + e^(−cy) = 1 with c = π/√6. So is the next one, and the next. A random partition of a large number has a shape, and the shape is known exactly.

number · Partitions One lattice, 3 generating sets, 3 shapes of ball. Lattice points reached within a fixed number of steps in the integers squared, for one step along either axis, axis steps and one diagonal, a king's moves, each drawn inside the polygon spanned by its steps and scaled by the radius.

The polygon a lattice becomes from far away

Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.

algebra · Cayley graph 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 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 A random tiling of the Aztec diamond, frozen outside a circle. A uniformly random domino tiling of the order-40 Aztec diamond (1640 dominoes) with the inscribed circle drawn.

The diamond that freezes its corners

A square board's domino tilings grow like 1.3385 choices per square. Arrange the same squares in a staircase-edged diamond and the count is exactly 2 to the n(n+1)/2 — only 1.1892 choices per square, at every size. The boundary has reached into the middle. Pick one of the diamond's tilings at random and the reason is visible: outside the circle inscribed in the diamond the dominoes are frozen into solid brickwork, and only inside it is the tiling free.

discrete · Halls theorem Three coordinates and a fourth power. Ball sizes at r = 10, 20, 44: Z² 221, 841, 3961; Z³ 1561, 11521, 117569; Heisenberg 4309, 68079, 1600703; local exponent at 30: Heisenberg 4.005, Z³ 2.946.

Three coordinates and a fourth power

The Heisenberg group is made of triples of whole numbers, and two moves generate all of it. Count the triples within r moves of the start and the count grows like r⁴, not r³: at forty-four moves there are 1,600,703 of them, against 117,569 for the cubic lattice. The fourth power comes from one coordinate that the moves reach by enclosing area, so that height costs only the square root of itself — and seen from far away, the ball of reachable triples is a curved solid with a dimple at each pole.

algebra · Cayley graph

Named alongside it

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

Cayley graphFluctuationsGrowth rateLatticeWord metricAbelian groupAsymptoticsBipartite graphCommutatorConjugate partitionConvex hullConvexity

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