Persistence exponents of self-interacting random walks

TL;DR

This paper provides exact calculations of the persistence exponent for self-interacting random walks, revealing their long-term survival probabilities and exploration characteristics in various scientific fields.

Contribution

It offers the first exact determination of the persistence exponent for all relevant self-interacting random walks, a class with complex memory effects.

Findings

Exact persistence exponents for all relevant SIRWs computed.

Splitting probabilities for SIRWs determined.

Results applicable to diverse fields like biology and machine learning.

Abstract

The persistence exponent, which characterises the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. Determining this exponent for non-Markovian processes is known to be a difficult task, and exact results remain scarce despite sustained efforts. In this Letter, we consider the fundamental class of self-interacting random walks (SIRWs), which display long-range memory effects that result from the interaction of the random walker at time $t$ with the territory already visited at earlier times $t'<t$. We compute exactly the persistence exponent for all physically relevant SIRWs. As a byproduct, we also determine the splitting probability of these processes. Besides their intrinsic theoretical interest, these results provide a quantitative characterization of the exploration process of SIRWs, which are involved in fields as diverse as foraging theory, cell biology, and machine learning.