Exploration and aging of non-Markovian processes

TL;DR

This thesis investigates how memory influences space exploration in non-Markovian random walks, revealing universal patterns, phase transitions, and exact results across various models, advancing understanding of memory effects in stochastic processes.

Contribution

It introduces a unified framework for memory effects in non-Markovian walks and provides the first exact results for aged observables in several models.

Findings

Identification of confinement criteria and phase transition in LARWs

Exact expressions for observables in SIRWs, including aging effects

Derivation of the first exact aged observable in TSAW

Abstract

This thesis explores a central question: how does memory affect the way random walkers explore space? By analyzing various non-Markovian models, where past behavior directly influences future dynamics, we uncover new mechanisms and universal patterns in space exploration. In the first part, we study Locally Activated Random Walks (LARWs), where motion depends on the time spent at visited sites. These walks, though simply defined, display rich behaviors such as dynamical trapping, aging, and non-Gaussian diffusion. We identify confinement criteria, reveal a dynamical phase transition, and derive exact analytical results in several regimes. The second part focuses on Self-Interacting Random Walks (SIRWs), where the walker's own history governs future steps. Using advanced probabilistic tools, especially Ray-Knight theory, we compute exact results for observables such as splitting probabilities, persistence, first-passage times, number of visited sites, and aging effects. This includes the first exact expressions for aged observables in the saturating Self-Attracting Walk (SATW) model. In the third part, we propose a unified framework to describe memory effects in space exploration, based on the walker's tendency to reverse direction over time. This approach reveals robust patterns across a wide class of non-Markovian models, including fractional Brownian motion and the strongly self-interacting True Self-Avoiding Walk (TSAW), where we derive the first exact result for an aged observable in a non-saturating system. Together, these results show that memory-driven systems, though complex, can exhibit universal structures. This work offers new tools to analyze such dynamics and lays the groundwork for future research across physics, probability, and beyond.