Convergence results and sharp estimates for the voter model interfaces
Samir Belhaouari, EPFL
Thomas Mountford, EPFL
Rongfeng Sun, EURANDOM
Glauco Valle, EPFL / DME-IM-UFRJ
Abstract
We study the evolution of the interface for the one-dimensional voter
model. We show that if the random walk kernel associated with the voter
model has finite γth moment for some γ>3, then the
evolution of the interface boundaries converge weakly to a Brownian
motion under diffusive scaling. This extends recent work of Newman,
Ravishankar and Sun. Our result is optimal in the sense that finite
γth moment is necessary for this convergence for all γ ∈ (0,3).
We also obtain relatively sharp estimates for the tail distribution of
the size of the equilibrium interface, extending earlier results of Cox
and Durrett, and Belhaouari, Mountford and Valle.
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