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Random walk with random resetting to the maximum position.

Satya N Majumdar1, Sanjib Sabhapandit2, Grégory Schehr1

  • 1LPTMS, CNRS, Univ Paris Sud, Université Paris-Saclay, 91405 Orsay, France.

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Summary
This summary is machine-generated.

This study analyzes a resetting random walk model. For a non-zero resetting probability (r>0), both the walker's average position and maximum visited site grow ballistically, with distinct crossover behaviors observed at r=0.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Stochastic Processes

Background:

  • Standard random walks exhibit diffusive behavior (distance ~√n).
  • Resetting mechanisms introduce non-trivial dynamics to stochastic processes.
  • Understanding anomalous diffusion is crucial in various scientific fields.

Purpose of the Study:

  • To analytically investigate a one-dimensional random walk model with a resetting mechanism.
  • To characterize the statistical properties of the walker's position and maximum visited site.
  • To explore the impact of the resetting probability (r) on growth laws and distributions.

Main Methods:

  • Analytical treatment of a discrete-time random walk model.
  • Calculation of average position, average maximum, and their fluctuations.
  • Analysis of probability distributions and large deviation functions.

Main Results:

  • For r>0, average position and maximum grow ballistically (speed v(r)).
  • Fluctuations grow diffusively with a common diffusion coefficient D(r).
  • A dynamical phase transition is observed with a weakly singular large deviation function.

Conclusions:

  • The resetting rate (r) fundamentally alters random walk behavior from diffusive to ballistic.
  • r=0 is a critical point, exhibiting different growth laws compared to the r→0 limit.
  • Exact crossover functions describe the transition between critical and off-critical regimes.