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 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.
be the maximal value of a Gaussian random function
on the (two-dimensional) sphere.
Assume that the expected value of
(at every point of the sphere), and the standard deviation of
(at every point of the sphere).
(the standard normal distribution), and
is a constant; it does not depend on
, but depends on the correlation function of
The relative error of the approximation decays exponentially for large
is easy to determine in the important special case described in terms of the directional derivative of
at a given point (of the sphere) in a given direction (tangential to the sphere).
The derivative is random, with zero expectation and some standard deviation.
The latter may depend on the point and the direction.
However, if it does not depend, then it is equal to
(for the sphere of radius
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 to the theory sketched above is, Euler characteristic
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 whenever
is non-empty; its Euler characteristic may take various values, depending on the topology of the set (the number of connected components, and possible holes in these components).
is usually a small, slightly deformed disk or ellipse (which is easy to guess, but quite difficult to prove).
Thus, its Euler characteristic
is usually equal to
The basic statement given above is a simple special case of a much more general (and difficult) theory stated by Adler.
[1][2][3] For a detailed presentation of this special case see Tsirelson's lectures.