Clustering in one-dimensional threshold voter models

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, ..., 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 [greater-or-equal, slanted] 1 in the sense that P[[eta]t(k) [not equal to] [eta]t(j)] tends to zero as t tends to [infinity] for every initial configuration. Furthermore, the limiting distribution as t --> [infinity] exists (and is a mixture of the pointmasses on [eta] [reverse not equivalent] 1 and [eta] [reverse not equivalent] 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([varrho]) 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, ], convex on [, 1], and has derivative 2 at 0. If N [greater-or-equal, slanted] 2, this derivative is zero.