Diffusive-Ballistic Transition in Random Walks with Long-Range Self-Repulsion

TL;DR

This paper proves that certain two-dimensional random walks with long-range self-repulsive interactions exhibit a phase transition between diffusive and ballistic behaviors.

Contribution

It establishes the existence of a phase transition in random walks with long-range self-repulsion, a novel result in the study of such stochastic processes.

Findings

Existence of a diffusive-ballistic phase transition

Characterization of the phase transition in long-range self-repulsive walks

Mathematical proof of the transition in $ ext{Z}^2$

Abstract

We prove that a class of random walks on $\Z^2$ with long-range self-repulsive interactions have a diffusive-ballistic phase transition.