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This study introduces an adaptive search model where random walkers reset to visited sites. A phase transition emerges, creating localized, non-diffusive states and demonstrating how non-Markovian movement aids environmental learning.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Random walks are fundamental models in physics, but often assume memoryless (Markovian) behavior.
  • Understanding adaptive search strategies is crucial for fields ranging from biology to computer science.
  • Previous models often overlook the role of memory and reinforcement in search dynamics.

Purpose of the Study:

  • To investigate an adaptive search model incorporating stochastic resetting to previously visited sites.
  • To identify phase transitions and emergent behaviors in a d-dimensional lattice with a trapping site.
  • To explore the implications of non-Markovian movement for environmental learning and search efficiency.

Main Methods:

  • Solving an adaptive search model with Lévy flight or random walker dynamics.
  • Introducing stochastic resetting to previously visited sites on a d-dimensional lattice.
  • Analyzing the emergence of non-diffusive stationary states and phase transitions.

Main Results:

  • A phase transition occurs at a critical resetting rate, leading to non-diffusive states localized around a trapping site.
  • The transition threshold depends on trapping strength and walker return probability.
  • The observed transition is analogous to Anderson localization, indicating a fundamental physical mechanism.

Conclusions:

  • Non-Markovian movement, characterized by resetting to visited sites, enables agents to learn about their environment.
  • Adaptive search strategies can outperform traditional diffusion-based methods in complex environments.
  • This model offers insights into biological search behaviors and potential applications in optimization and robotics.