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Extremal statistics for stochastic resetting systems.

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Extreme value statistics (EVS) of stochastic resetting systems reveal that the time of maximum displacement becomes uniformly distributed, independent of the underlying process at large times. This finding is crucial for understanding rare events in diverse scientific fields.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Stochastic Processes

Background:

  • Non-equilibrium systems analysis often relies on averages, but rare events and extreme fluctuations are critical for a complete understanding.
  • Extreme value statistics (EVS) are vital across disciplines like physics, finance, and ecology for analyzing significant deviations.
  • Stochastic resetting systems, with applications in search, physics, and computer science, are increasingly studied for their unique dynamics.

Purpose of the Study:

  • To investigate the extreme value statistics (EVS) of spatial displacement in stochastic resetting systems.
  • To analyze the finite and large time statistics of maximum displacement and the time it occurs (arg-maximum).
  • To establish a relationship between the joint distribution of maximum and arg-maximum and the underlying process characteristics.

Main Methods:

  • Derivation of an exact renewal formula connecting the joint distribution of maximum and arg-maximum to underlying process statistics.
  • Benchmarking results for the maximum against the Gumbel distribution for large sample sizes.
  • Analysis of various stochastic processes including diffusion, diffusion with drift, Ornstein-Uhlenbeck, and random acceleration processes under resetting.

Main Results:

  • The maximum of a reset trajectory follows the Gumbel class for large sample sizes.
  • The arg-maximum density converges to a uniform distribution at large observation times, independent of the specific underlying process.
  • This uniform distribution of the arg-maximum is a direct consequence of the renewal property introduced by the resetting mechanism.

Conclusions:

  • The study provides a comprehensive analysis of extreme value statistics in stochastic resetting systems.
  • The emergence of a uniform distribution for the arg-maximum highlights a universal feature of resetting mechanisms.
  • These findings offer valuable insights into the behavior of systems dominated by rare events and extreme fluctuations.