Long-range voter model on the real line
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Abstract
In the classical voter model on the integer lattice, a voter with one of two opinions is placed at each location on the lattice. Each voter has an alarm clock set to an exponentially distributed random time. When the clock rings, the voter adopts the opinion of a randomly chosen neighbour and the clock is set at a new random time. This process satisfies a moment duality with a system of coalescing random walks.
Here we are interested in the situation with an uncountable number of voters, placed at each point of the real line. We allow them to adopt opinions of other voters that are far away. Specifically, we describe a measure valued process satisfying a moment duality relation with a coalescing system of symmetric α-stable processes, where α ∈ (1, 2). Such a process was constructed by Steven N. Evans in 1997.
In this thesis we discuss the Hausdorff dimension of the interface between opinions, the survival probability at a given time point and the dimension of the time points where the support is unbounded in a simplified model. Furthermore, we show that the process arises as limit of solutions of a stochastic partial differential equation with accelerated noise.