Tightness of voter model interfaces

TL;DR

This paper proves that one-dimensional long-range voter models exhibit interface tightness under certain conditions, extending previous results and introducing a new proof method, including for a mixed long-range swapping voter model.

Contribution

It provides a new, concise proof that finite second moment infection rates ensure interface tightness and extends the result to a long-range swapping voter model.

Findings

Finite second moment suffices for interface tightness

New short proof of existing tightness result

Interface tightness shown for a long-range swapping voter model

Abstract

Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.