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Marie Chupeau1, Olivier Bénichou1, Raphaël Voituriez2

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Area of Science:

  • Statistical Physics
  • Probability Theory
  • Complex Systems

Background:

  • Cover time is crucial for understanding exploration dynamics in confined systems.
  • Persistent random walks offer a simple yet powerful model for systems with short-range memory.
  • Existing models often lack exact solutions for mean cover time in 1D lattices.

Purpose of the Study:

  • To derive the exact mathematical expression for the mean cover time of a persistent random walk on a one-dimensional lattice.
  • To analyze the impact of boundary conditions (periodic and reflecting) on the mean cover time.
  • To provide a foundational analytical tool for studying memory effects in random walk processes.

Main Methods:

  • Analytical derivation using techniques from stochastic processes.
  • Formulation of the persistent random walk model on a discrete 1D lattice.
  • Calculation of mean first passage times and cover times under different boundary conditions.

Main Results:

  • An exact analytical expression for the mean cover time of a persistent random walker on a 1D lattice was obtained.
  • The derived formula explicitly accounts for the walker's memory (persistence).
  • Distinct solutions were found for periodic and reflecting boundary conditions, highlighting their influence.

Conclusions:

  • The study provides a precise, closed-form solution for mean cover time in a minimal persistent random walk model.
  • This work offers valuable insights into the role of short-range memory in diffusion and exploration processes.
  • The derived formulas serve as a benchmark for more complex models and simulations in statistical physics and related fields.