Interfaces of the one-dimensional voter models

We show that for a voter model on {0,1}Z corresponding to a random walk with kernel p(·) 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(·) has second moments but does not if p(·) fails to have α moment for some α < 2. 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 converges weakly to a Brownian motion. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that a 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.

Mountford, Thomas
Lausanne, EPFL
Other identifiers:
urn: urn:nbn:ch:bel-epfl-thesis3614-6

Note: The status of this file is: EPFL only

 Record created 2006-06-21, last modified 2018-03-17

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