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's and 1's exists if $p(\cdot )$ has finite second moment but does not if $p(\cdot )$ fails to have finite moment of order $\alpha$ for some $\alpha <2$.

20 pages