A random function of either one variable (a random process), or two or more variables (a random field) is called Gaussian if every finite-dimensional distribution is a multivariate normal distribution. Gaussian random fields on the sphere are useful (for example) when analysing

• the anomalies in the cosmic microwave background radiation (see, pp.89);

• brain images obtained by positron emission tomography (see, pp.910).

Sometimes, a value of a Gaussian random function deviates from its expected value by several standard deviations. This is a large deviation. Though rare in a small domain (of space or/and time), large deviations may be quite usual in a large domain.

Basic statement

(the standard normal distribution, but depends on the correlation function (see below). The relative error.

is easy to determine in the important special case described in terms of the directional derivative at a given point (of the sphere) in a given direction (tangential).

is in fact the Euler characteristic of the sphere (for the torus it vanishes).

is twice continuously differentiable (almost surely), and reaches its maximum at a single point (almost surely).

The clue: mean Euler characteristic

of the set can be calculated explicitly:

(which is far from being trivial, and involves Poincaré–Hopf theorem, Gauss–Bonnet theorem, Rice's formula etc.).

is the empty set is non-empty; its Euler characteristic may take various values, depending on the topology of the set (the number of connected components is usually a small, slightly deformed disk or ellipse.

See also

• Gaussian process

• Gaussian random field

• Large deviations theory

Further reading

The basic statement given above is a simple special case of a much more general (and difficult) theory stated by Adler. For a detailed presentation of this special case see Tsirelson's lectures.