The Single Big Jump Principle in Physical Modelling

Abstract

The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell.

23 pages, 11 figures

References in corpus (16)

• The large deviation approach to statistical mechanics
• Lévy walks
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Universal Record Statistics of Random Walks and Lévy Flights
• On the influence of time and space correlations on the next earthquake magnitude
• Infinite densities for Lévy walks
• Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics
• Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution
• Non-homogeneous persistent random walks and averaged environment for the Lévy-Lorentz gas
• Development and regression of a large fluctuation
• Anomalous transmission and drifts in one-dimensional Levy structures
• Strong anomalous diffusion of the phase of a chaotic pendulum
• Heat perturbations spreading in the Fermi-Pasta-Ulam-$\beta$ system with next-nearest-neighbor coupling: Competition between phonon dispersion and nonlinearity
• Equivalence of position-position auto-correlations in the Slicer Map and the Lévy-Lorentz gas
• Continuous-time random walk between Lévy-spaced targets in the real line

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• Limit theorems for Lévy flights on a 1D Lévy random medium
• Data-driven analysis of annual rain distributions
• Large-time asymptotic of heavy tailed renewal processes
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• In the folds of the Central Limit Theorem: Lévy walks, large deviations and higher-order anomalous diffusion
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• Generalized autocorrelation function in the family of deterministic and stochastic anomalous diffusion processes
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• Anatomy of an extreme event: What can we infer about the history of a heavy-tailed random walk?
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value
• Big jump principle for heavy-tailed random walks with correlated increments
• First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension
• Mean first passage times of higher-dimensional velocity jump processes
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