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Tightness for the interfaces of one-dimensional voter models. (English) Zbl 1112.60074
Summary: We show that for the voter model on ${\left\{0,1\right\}}^{ℤ}$ corresponding to a random walk with kernel $p\left(·\right)$ and starting from unanimity to the right and opposing unanimity to the left, a tight interface between zeros and ones exists if $p\left(·\right)$ has finite second moment but does not if $p\left(·\right)$ fails to have finite moment of order $\alpha$ for some $\alpha <2$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F17 Functional limit theorems; invariance principles 82B41 Random walks, random surfaces, lattice animals, etc. (statistical mechanics) 82B24 Interface problems; diffusion-limited aggregation (equilibrium statistical mechanics)