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  • 1Istituto dei Materiali per l'Elettronica ed il Magnetismo (IMEM-CNR), Parco Area delle Scienze, 37/A-43124 Parma, Italy; Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy; and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy.

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This study introduces a method to estimate rare event probabilities in jump processes with fat-tailed distributions. Fast, rare events can significantly impact outcomes, even on much shorter timescales than typical process durations.

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Area of Science:

  • * Physics, Mathematics, and Computational Science
  • * Statistical Mechanics and Probability Theory

Background:

  • * Rare events in first-passage time distributions can initiate anomalous reactions.
  • * Accurate estimation of rare event probabilities is crucial for processes with broad-tailed jump distributions, where rare events occur more frequently.
  • * Jump processes are widely used to model diverse phenomena across various scientific disciplines.

Purpose of the Study:

  • * To develop a general framework for quantifying the contribution of fast rare events to exit probabilities in systems with fat-tailed distributions.
  • * To analyze the impact of these rare events on specific jump processes, including discrete time random walks, Lévy walks, and the Lévy-Lorentz gas.
  • * To determine the scaling function governing the probability distribution of fast rare events leading to early exit at a distant point.

Main Methods:

  • * Formulation of a general approach to estimate the contribution of fast rare events to exit probabilities.
  • * Application of this approach to analyze three distinct jump processes: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas.
  • * Derivation of the exact scaling function for the probability distribution of fast rare events.

Main Results:

  • * Identified that events on timescales significantly shorter than the typical process timescale can substantially influence exit probabilities.
  • * Determined the precise scaling function for the probability distribution of fast rare events in the studied jump processes.
  • * Demonstrated that these fast rare events play a critical role in the process exiting an interval rapidly and at a large distance.

Conclusions:

  • * The developed approach provides a robust method for estimating rare event probabilities in systems with fat-tailed distributions.
  • * Fast rare events are shown to be significant contributors to the overall exit probability, challenging assumptions based on typical timescales.
  • * The findings have broad implications for modeling and understanding phenomena in fields such as biology, transport, ecology, and finance.