Universal power-law tails for singularity-dominated strong fluctuations

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Abstract
Integrals of the type I(t) identical to integral integral dxdy delta (t-H(x,y)) are considered, where H is a function from an ensemble labelled by parameters. I can represent circulation times of fluid particles in the plane, orbital periods or semiclassical densities of states for one-dimensional Hamiltonian systems, spectral densities for two-dimensional crystals, or the strength of a wave pulse produced by propagation of a deformed step discontinuity. As H varies over the ensemble, I develops strong fluctuations associated with Legendre singularities. For unrestricted functions H, the probability distribution P(I) describing the fluctuations of I decays according to a universal I-9 law, obtained by a scaling argument involving Arnold's classification of catastrophes. If H is only quadratic in y, P varies as I-10. If the integral defining I is one-dimensional, P varies as I-3; if it is three-dimensional, I decays no faster than I-48. The probability distribution of I' identical to mod delta I/ delta t mod decays as (I')-2.

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