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.