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Clustering in one-dimensional threshold voter models. (English) Zbl 0752.60086
Summary: We consider one-dimensional spin systems in which the transition rate is 1 at site \(k\) if there are at least \(N\) sites in \(\{k-N,k-N+1,\dots,k+N- 1,k+N\}\) at which the ‘opinion’ differs from that at \(k\), and the rate is zero otherwise. We prove that clustering occurs for all \(N\geq 1\) in the sense that \(P[\eta_ t(k)\neq\eta_ t(j)]\) tends to zero as \(t\) tends to \(\infty\) for every initial configuration. Furthermore, the limiting distribution as \(t\to \infty\) exists (and is a mixture of the pointmasses on \(\eta\equiv 1\) and \(\eta\equiv 0\)) if the initial distribution is translation invariant. In case \(N=1\), the first of these results was proved and a special case of the second was conjectured in a recent paper by J. T. Cox and R. Durrett [Random walks, Brownian motion, and interacting particle systems, Festschr. in Honor of Frank Spitzer, Prog. Probab. 28, 189-201 (1991)]. Now let \(D(\rho)\) be the limiting density of 1’s when the initial distribution is the product measure with density \(\rho\). If \(N=1\), we show that \(D(\rho)\) is concave on \([0,{1\over 2}]\), convex on \([{1\over 2},1]\), and has derivative 2 at 0. If \(N\geq 2\), this derivative is zero.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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