Large deviations of Gaussian random functions
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. Basic statement. Let formula_1 be the maximal value  of a Gaussian random function formula_2 on the (two-dimensional) sphere. Assume that the expected value of formula_2 is formula_4 (at every point of the sphere), and the standard deviation of formula_2 is formula_6 (at every point of the sphere). Then, for large formula_7, formula_8 is close to formula_9, where formula_10 is distributed formula_11 (the standard normal distribution), and formula_12 is a constant; it does not depend on formula_13, but depends on the correlation function of formula_2 (see below). The relative error of the approximation decays exponentially for large formula_13. The constant formula_12 is easy to determine in the important  special case described in terms of the directional derivative of formula_2 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 formula_18 (for the sphere of radius formula_6). The coefficient formula_20 before formula_21 is in fact the Euler characteristic of the sphere (for the torus it vanishes). It is assumed that formula_2 is twice continuously differentiable (almost surely), and reaches its maximum at  a single point (almost surely). The clue: mean Euler characteristic. The clue to the theory sketched above is, Euler characteristic formula_23 of the set formula_24 of all points formula_25 (of the sphere) such that formula_26. Its expected value (in other words, mean value) formula_27 can be calculated explicitly: (which is far from being trivial, and involves Poincaré–Hopf theorem, Gauss–Bonnet theorem, Rice's formula etc.). The set formula_24 is the empty set whenever formula_30. In the other case, when formula_31, the set formula_24 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). However, if formula_13 is large and formula_31 then the set formula_24 is usually a small, slightly deformed disk or ellipse (which is easy to guess, but quite difficult to prove). Thus, its Euler characteristic formula_23 is usually equal to formula_6 (given that formula_31). This is why formula_39 is close to formula_8. 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. Large deviations