@article{EJP6,
author = {Klaus Fleischmann and Andreas Greven},
title = {Time-Space Analysis of the Cluster-Formation in Interacting Diffusions},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {1},
year = {1996},
keywords = {interacting diffusion, clustering, infinite particle system, delayed coalescing random walk with immigration, transformed Fisher-Wright tree, low dimensional systems, ensemble of log-coalescents},
abstract = {A countable system of linearly interacting diffusions on the interval [0,1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).},
pages = {no. 6, 1-46},
issn = {1083-6489},
doi = {10.1214/EJP.v1-6},
url = {http://ejp.ejpecp.org/article/view/6}}