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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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##### References:
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