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

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

Named alongside it

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

Asymptotics Cayley graph Conjugate partition Convex hull Durfee square Ends of a group Ferrers diagram Generating function Growth rate Lattice Norm Partition

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