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Large deviations in continuous-time random walks.

Adrian Pacheco-Pozo1, Igor M Sokolov2

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Summary
This summary is machine-generated.

This study explores large deviation properties of continuous-time random walks (CTRWs). We provide a general formula for CTRW large deviation rates, simplifying for Gaussian step distributions.

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Area of Science:

  • Physics
  • Mathematics
  • Probability Theory

Background:

  • Continuous-time random walks (CTRWs) are fundamental models in statistical physics and probability theory.
  • Understanding the large deviation properties of CTRWs is crucial for analyzing anomalous transport phenomena.

Purpose of the Study:

  • To derive a general expression for the large deviation rate in CTRWs.
  • To investigate the relationship between large deviation rates and the underlying distributions of step lengths and waiting times.

Main Methods:

  • Derivation of a general formula for large deviation rates in CTRWs.
  • Application of Legendre transformations to the cumulant generating function for Gaussian step distributions.
  • Analysis of specific examples including Bernoulli and Gaussian random walks with various waiting time distributions (exponential, Lévy, Pareto).

Main Results:

  • A general expression for the large deviation rate in CTRWs is presented.
  • For Gaussian step lengths, the rate simplifies to two Legendre transformations of the waiting time cumulant generating function.
  • Analysis of diverse examples reveals general properties of large deviations in CTRWs.

Conclusions:

  • The derived general expression offers a unified framework for studying large deviations in CTRWs.
  • The findings highlight the significant impact of waiting time distributions on large deviation behavior.
  • The study provides insights into the statistical properties of CTRWs under extreme conditions.