Myopic non-intersection in a periodic potential

TL;DR

This paper studies a class of Markov processes called myopic non-intersecting Brownian motions under a periodic potential, analyzing their long-term behavior and introducing an explicit sampling algorithm for such constrained systems.

Contribution

It introduces a new class of Markov processes with myopic non-intersection constraints and develops an explicit acceptance-rejection sampling algorithm for their construction.

Findings

Long-term convergence to non-intersecting random walks

Transition between non-intersection and exclusion dynamics

Effective sampling algorithm for constrained systems

Abstract

We introduce a class of Markov processes conditioned to avoid intersection over a moving time window of length T>0, a setting we refer to as myopic non-intersection. In particular, we study a system of myopic non-intersecting Brownian motions subject to a periodic potential. Our focus lies in understanding the interplay between the confining effect of the potential and the repulsion induced by the non-intersection constraint. We show that, in the long time limit, and as both T and the strength of the potential become large, the model converges to a system of myopic non-intersecting random walks, which transitions between standard non-intersection dynamics and exclusion behavior. The main technical contribution of the paper is the introduction of an algorithm, based on a modification of the acceptance-rejection sampling scheme, that provides an explicit construction of myopically constrained systems.