Exact Propagators of One-Dimensional Self-Interacting Random Walks

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.