Abstract
Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.
Highlights
Introduction and main resultsLet X = (Xt)t≥0 be a long-range, one-dimensional ‘swapping’ voter model, i.e. X is a Markov process with state space {0, 1}Z and formal generator G := Gv + Gs, whereGvf (x) := q(i − j)1{x(i)=x(j)}{f (x{i}) − f (x)}, ij Gsf (x) :=1 2 p(i − j)1{x(i)=x(j)}{f (x{i,j}) − f (x)}, ij (1.1)Electronic Communications in Probability q and p are functions Z → [0, ∞), and p is symmetric, i.e., p(i) = p(−i) (i ∈ Z)
If the process obtained by identifying states that are translations of each other is positively recurrent, it is said that the voter model exhibits interface tightness
In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment
Summary
If the process obtained by identifying states that are translations of each other is positively recurrent, it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Let X = (Xt)t≥0 be a long-range, one-dimensional ‘swapping’ voter model, i.e. X is a Markov process with state space {0, 1}Z and formal generator G := Gv + Gs, where
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