Ballistic Phase of Self-Interacting Random Walks

TL;DR

This paper presents a unified theoretical framework using Ornstein-Zernike theory to analyze the ballistic phase in various self-interacting random walks and polymers, providing local limit results across different deviation scales.

Contribution

It introduces a comprehensive approach to study the ballistic phase in a broad class of self-interacting models, extending previous methods with new local limit results.

Findings

Established local limit theorems for displacement and pattern hits.

Unified analysis applicable to multiple models including self-avoiding walks and reinforced polymers.

Demonstrated the universality of the ballistic phase across different models.

Abstract

We explain a unified approach to a study of ballistic phase for a large family of self-interacting random walks with a drift and self-interacting polymers with an external stretching force. The approach is based on a recent version of the Ornstein-Zernike theory developed in earlier works. It leads to local limit results for various observables (e.g. displacement of the end-point or number of hits of a fixed finite pattern) on paths of n-step walks (polymers) on all possible deviation scales from CLT to LD. The class of models, which display ballistic phase in the "universality class" discussed in the paper, includes self-avoiding walks, Domb-Joyce model, random walks in an annealed random potential, reinforced polymers and weakly reinforced random walks.