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Survival-time statistics for sample space reducing stochastic processes.

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This study introduces a solvable stochastic process with a shrinking state space, revealing Gaussian survival-time distributions that scale logarithmically with system size. This model connects to record statistics in complex systems.

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Area of Science:

  • Complex Systems
  • Stochastic Processes
  • Statistical Mechanics

Background:

  • Complex systems often exhibit scale-invariant features.
  • Stochastic processes with time-varying state spaces can model these phenomena.

Purpose of the Study:

  • To analyze a sample-space reducing (SSR) stochastic process.
  • To evaluate the survival-time probability distribution of this model.
  • To explore connections between SSR processes and record statistics.

Main Methods:

  • Defined an SSR stochastic process generating strictly decreasing integer sequences.
  • Analytically evaluated the exact survival probability P_N(τ).
  • Investigated the asymptotic behavior for large system sizes (N).

Main Results:

  • The survival probability distribution approaches a Gaussian form for large N.
  • The mean survival time and its variance scale logarithmically with system size (〈τ〉∼lnN, σ_τ^2∼lnN).
  • Established a correspondence between SSR survival-time statistics and record statistics.

Conclusions:

  • The SSR model provides an exactly solvable framework for scale-invariant phenomena.
  • Logarithmic scaling of mean and variance is a key characteristic of this process.
  • The model offers insights into the statistical properties of complex systems and random variables.