Dynamics and Self-Similarity in Min-Driven Clustering

Govind Menon Division of Applied Mathematics Box F Brown University Providence, RI 02912 Email: menon@dam.brown.edu

Barbara Niethammer Mathematical Institute University of Oxford Oxford, OX1 3LB, UK Email: niethammer@maths.ox.ac.uk

and

Robert L. Pego Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 Email: rpego@cmu.edu

Abstract: We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer k is chosen with probability p_{k}, and the smallest cluster merges with k randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in (0, ∞), and we establish necessary and sufficient conditions for approach to self-similar form. We also characterize eternal solutions for this model via a Lévy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale.