Clustering and dispersion rates for some interacting particle systems on $Z$
Ann. Probab., 8 (2):
183--213 (1980)Abstract
It is well known that the votermodel on Zd under initialproductmeasure
will convergeweaklyto a pointprocess as t -+ oo; ifd > 3, convergence wil be
to a nontrivialpoint process; if d = 1, 2, convergence will be to a linear
combinationof trivialpoint processes. Therefore, d = 1, 2, the clustersize
for
large as
of a particularstate around any fixedpoint tends to become arbitrarily
t oo. Here we examine the rate of growth d = 1 of this clustering the
for for
nearestneighborvotermodel and the relatedproblem of interparticle distance
fornearestneighborcoalescing random walks and annihilating random walks.
We show that under spatial renormalization these clustersizes/interparticle
t2
distances in each case approach a nondegeneratedistribution. We examine
these distributions and obtain numericalestimatesfor these and related prob-
lems.
MR This paper deals with the limiting behavior of three one-dimensional interacting particle systems: the voter model, the coalescing random walk process, and the annihilating random walk process. Each is a Markov process on $\0,1\^Z$0,1Z. In the voter model, the öpinion'' at a site $xZ$x∈Z changes at an exponential rate which is proportional to the number of neighbors of $x$x which hold the opposite opinion. In the other two processes, particles undergo independent, continuous-time random walks with colliding particles coalescing in one process and annihilating each other in the other process. The limiting behavior of the distribution of each of these processes is elementary. If $\mu_0$μ0 and $\mu_1$μ1 denote, respectively, the point masses on äll zeros'' and äll ones'', then the distributions of the random walk processes converge to $\mu_0$μ0 for all initial states, while the distribution of the voter model converges to a mixture of $\mu_0$μ0 and $\mu_1$μ1 for any translation invariant initial distribution. Thus, clustering occurs in the voter model and dispersion occurs in the random walk process. The authors analyze the rates at which these occur. It is proved for appropriate initial distributions that, when divided by $t$√t, the (spatial) mean cluster size for the voter model and the (spatial) mean interparticle distances for the random walk processes converge to constants. Convergence in distribution is also obtained for the normalized size of the cluster containing the origin for the voter model and for the normalized distances between the two particles on either side of the origin for the random walk processes. The proofs rely, among other things, on the duality relations which hold for these processes.
Description
MR: Publications results for "MR Number=(566588)"