Concept

Random field

A random function of position, such as a landscape whose height at each point is a random variable correlated with the heights nearby. Gaussian random fields are fixed by their covariance, and averages of local features, like the Euler characteristic of a level set, can be computed exactly.

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

A random surface and the region above height 1. A smooth Gaussian random field on a 64 by 64 torus and its excursion set above 1: 14 pieces, 0 holes, Euler characteristic 14; expected 13.8.

Pieces minus holes on a random surface

Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.

topology · Euler characteristic

The pieces of a random vibration. Random plane wave, 240×240 grid, wavelength 24: 23 positive and 18 negative domains.

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.

topology · Euler characteristic

Named alongside it

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

Critical point Euler characteristic Saddle point Morse theory Percolation Scaling Torus

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