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We derived the propagator for two types of self-interacting random walks (SIRWs), revealing long-range memory effects. This breakthrough provides key insights into non-Markovian processes and their applications.

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Area of Science:

  • Physics
  • Mathematics
  • Statistical Mechanics

Background:

  • Self-interacting random walks (SIRWs) exhibit long-range memory, crucial for modeling phenomena like foraging and cell behavior.
  • These non-Markovian processes are challenging to analyze, with their fundamental propagator remaining elusive for most classes.
  • Existing theoretical frameworks struggle to fully characterize SIRW dynamics due to their inherent memory effects.

Purpose of the Study:

  • To derive an exact and explicit expression for the propagator of two significant SIRW universality classes.
  • To enable the calculation of key observables, such as the diffusion coefficient, for these complex random walks.
  • To elucidate the underlying non-Markovian mechanisms governing SIRW behavior.

Main Methods:

  • Analytical derivation of the propagator for once-reinforced random walks.
  • Exact calculation of the propagator for polynomially self-repelling walks.
  • Utilizing theoretical frameworks to analyze the statistical properties of SIRWs.

Main Results:

  • An explicit formula for the propagator of once-reinforced and polynomially self-repelling walks has been established.
  • Previously inaccessible observables, including the diffusion coefficient, can now be determined.
  • A novel non-Markovian mechanism driving walkers away from their origin was uncovered.

Conclusions:

  • The derived propagators provide a fundamental tool for understanding complex SIRW dynamics.
  • These findings advance the theoretical treatment of non-Markovian processes with memory effects.
  • The study opens new avenues for applying SIRW models in diverse scientific and computational fields.