Diffusive clustering in an infinite system of hierarchically interacting diffusions

Abstract

We study a system of linearly interacting diffusions on the interval [0,1], indexed by an infinite hierachical group with parameter N, modeling the evolution of gene frequencies accounting for migration and resampling. Fisher-Wright diffusions are the typical example for the diffusion term. A particular choice of the migration term guaranties that we are in the so-called diffusive clustering regime. The processes in the whole class considered and starting with a shift-ergodic initial law have qualitative properties just as in the Fisher-Wright case (universality). Also, the initial law plays here no role. Clusters of components with values either close to 0 or close to 1 grow on various different scales (diffusive clustering). More precisely, at time N"t#->##infinity# (spatial) cluster sizes measured in a hierarchical distance grow like #alpha#t (i.e. consist of N"#alpha#"t components) where 1-#alpha# element of (0,1) is the hitting time of the traps (0,1) for a time transformed Fisher-Wright diffusion. Nevertheless, the single components oscillate infinitely often between values close to 0 and close to 1, but in such a way that they spend fraction one of their time together close to the boundary. Diffusive clustering phenomena we believe are in fact common to many interacting systems and were first discussed by Cox and Griffeath (1986) for the planar simple voter model. On the way, we prove some scaling results on a specific coalescing random walk with delay on the hierarchical group. (orig.)Available from TIB Hannover: RR 5549(23)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman