  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.
