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 Cox and Durrett.
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 \( [0,12] \), convex on \( [12,1] \), and has derivative 2 at 0. If \( N\geq 2 \), this derivative is zero.
