Stuck Walks

TL;DR

This paper studies a class of self-interacting random walks on the integer line, showing that under certain conditions they become confined to finite intervals due to the interplay of attraction and repulsion effects.

Contribution

It provides a rigorous analysis of the asymptotic behavior of self-interacting walks, identifying conditions for confinement based on model parameters.

Findings

Random walks can be confined to finite intervals depending on parameters.

Self-attracting effects dominate in certain parameter regimes.

The model demonstrates the balance between attraction and repulsion influences.

Abstract

We investigate the asymptotic behaviour of a class of self-interacting nearest neighbour random walks on the one-dimensional integer lattice which are pushed by a particular linear combination of their own local time on edges in the neighbourhood of their current position. We prove that in a range of the relevant parameter of the model such random walkers can be eventually confined to a finite interval of length depending on the parameter value. The phenomenon arises as a result of competing self-attracting and self-repelling effects where in the named parameter range the former wins.