Tightness for the interfaces of one-dimensional voter models

Abstract

We show that for the voter model on $\{0,1\}^{\mathbb{Z}}$ corresponding to a random walk with kernel $p(\cdot)$ and starting from unanimity to the right and opposing unanimity to the left, a tight interface between 0&#39;s and 1&#39;s exists if $p(\cdot)$ has finite second moment but does not if $p(\cdot)$ fails to have finite moment of order $α$ for some $α<2$.