Exact propagators of one-dimensional self-interacting random walks

TL;DR

This paper derives exact propagators for two classes of one-dimensional self-interacting random walks, enabling precise analysis of their behavior and revealing mechanisms influencing their movement.

Contribution

It provides the first explicit formulas for the propagators of the once-reinforced and polynomially self-repelling walks, advancing understanding of non-Markovian processes.

Findings

Exact propagator formulas for specific SIRWs

Determination of diffusion coefficients for these walks

Identification of non-Markovian mechanisms affecting walk dynamics

Abstract

Self-interacting random walks (SIRWs) show long-range memory effects that result from the interaction of the random walker at time $t$ with the territory already visited at earlier times $t'<t$. This class of non-Markovian random walks has applications in contexts as diverse as foraging theory, the behaviour of living cells, and even machine learning. Despite this importance and numerous theoretical efforts, the propagator, which is the distribution of the walker's position and arguably the most fundamental quantity to characterize the process, has so far remained out of reach for all but a single class of SIRW. Here we fill this gap and provide an exact and explicit expression for the propagator of two important classes of SIRWs, namely, the once-reinforced random walk and the polynomially self-repelling walk. These results give access to key observables, such as the diffusion coefficient, which so far had not been determined. We also uncover an inherently non-Markovian mechanism that tends to drive the walker away from its starting point.